text:
\section{Introduction}
An important topic of interest, in modeling of saturated flow through porous media, is the analysis of the phenomenon when there is a microstructure present in the rock matrix. Some of the achievements in the preexisting literature address the case when the microstructure is periodic; see \cite{ShowalterMicro, ShowWalk90, ShowWalk91} for the analytical approach and \cite{ArbBrunson2007, ArbLehr2006} for the numerical point of view. From a different perspective, in  \cite{Morales3} is presented the homogenization analysis for a non-periodic fissured system where the geometry of the cracks' surface satisfies $C^{1}$-smoothness hypotheses. However, none of these mathematical analysis accomplishments, takes in consideration a fractal geometric structure of porous media, which is an important case due to the remarkable evidence of this fact; in particular in \cite{KatzThompson} the authors found that pore space and pore interface have fractal features. Also, see \cite{ZhengShiChen} for pore structure characterization, including random growth models.  See  \cite{FederJosang} for fractal geometry results in tracing experiments within a porous medium including dispersion, fingering and percolation. Finally, \cite{ZhangLiTangGuo} discusses the use of fractal surfaces in modeling the storage phenomenon in gas reservoirs.

This paper concentrates on finding adequate fluid transmission conditions for flow in porous media with fractal interface, as well as the well-posedness of the corresponding weak variational formulations. Our goal is to ``blend" the modeling of porous media flow with the fractal roughness of the microstructure. We differ from the previously mentioned achievements since we replace the geometric feature of periodicity by that of self-similarity and explore, numerically, the effect of some randomness in the fractal geometry. On the other hand, the present study has a very different approach to the analysis on fractals from the preexisting literature, given that the mainstream PDE analysis on fractals concentrates its efforts in solving strong forms on the fractal domain \cite{Strichartz3}, the analysis of the associated eigenvalues and eigenfunctions \cite{BarlowKigami}, or the study of the adequate function spaces \cite{Strichartz1}.

In order to start understanding the key features of the phenomenon, we limit the study to the 1-D setting, defining the domain of analysis as $\Omega\defining(0,1)$. Additionally, we will use and adjustment of the classic stationary  diffusion problem \eqref{Eqn Darcy Classic} below, in order to introduce the fluid exchange transmission conditions across the fractal interface
\begin{align}\label{Eqn Darcy Classic}
& -\boldsymbol{\partial }  \big(K\, \boldsymbol{\partial }  p\big) = F &
& \text{in }\, \Omega \, , &
& p(0) = 0 \, ,    &
& \boldsymbol{\partial }  p (1) = 0\, .
\end{align}
Here $p$ stands for the pressure, $K\boldsymbol{\partial }  p$ indicates the flux according to Darcy's law and $K$ denotes de permeability, which will be set as $K = 1$ throughout this work. In addition, we set Dirichlet and Neumann boundary conditions on the extremes of the interval. We use classical notation and results on function spaces $L^{2}(\Omega), H^{1}(\Omega)$ and indicate with standard letters $p, q, r, u, v$ the functions on these spaces. We adopt the letters $\mathsf{B} $, $\mathpzc{b} $ to denote the microstructure and its elements respectively, in particular, the following classical Hilbert space will be frequently used
\begin{subequations}\label{Eqn Hilbert Space on Fractal}
\begin{equation}\label{Eqn Functions Hilbert Space on Fractal}
\ell^{\,2}(\mathsf{B} )\defining \Big\{g: \mathsf{B} \rightarrow \boldsymbol{\mathbbm{R} } \, \big\vert \sum\limits_{\mathpzc{b} \, \in\, \mathsf{B} }\vert g(\mathpzc{b} ) \vert^{2} < +\infty\Big\}\, ,
\end{equation}
endowed with its natural inner product
\begin{equation}\label{Eqn Inner Product Hilbert Space on Fractal}
\big\langle g, h \big\rangle_{\ell^{2}(\mathsf{B} )} \defining \sum\limits_{\mathpzc{b} \, \in\, \mathsf{B} } g(\mathpzc{b} ) \, h(\mathpzc{b} ).
\end{equation}
\end{subequations}
In the next section, we introduce the geometry of the fractal interface, together with the adequate mathematical setting in order to include it successfully in the PDE model.
\subsection{Geometric Setting}
Throughout this work we limit to a particular type of fractal microstructure, first we need to introduce a previous definition and a related result (see \cite{Falconer})
\begin{definition}[Iterated Function Systems]
Let $D\subseteq \boldsymbol{\mathbbm{R} } ^{\! N}$ be a closed set
\begin{enumerate}[(i)]
\item A function $S: D\rightarrow D$ is said to be a \textbf{contraction} if there exists a constant $c\in[0, 1)$ such that $\big\vert S(x) - S(y) \big\vert\leq c \, \big\vert x - y \big\vert$ for all $x, y\in D$. Additionally, we say that $S$ is a \textbf{similarity} if $\big\vert S(x) - S(y) \big\vert = c \, \big\vert x - y \big\vert$ for all $x, y\in D$ and $c$ is said to be its \textbf{ratio}.

\item A finite family of contractions, $\{S_{1}, S_{2}, \ldots, S_{L} \}$ on $D$ with $L\geq 2$ is said to be an \textbf{iterated function system} or \textbf{IFS}.

\item A non-empty compact subset $F\subseteq D$ is said to be an \textbf{attractor} of the IFS $\{S_{1}, S_{2}, \ldots, S_{L} \}$ if
\begin{equation}\label{Def Attractor of an IFS}
F = \bigcup_{i \, = \, 1}^{L}S_{i}(F)\, .
\end{equation}
In particular, if every contraction of the IFS is a similarity then, the attractor $F$ is said to be a \textbf{strictly self-similar} set.
\end{enumerate}
\end{definition}
\begin{theorem}
Consider the IFS given by the contractions $\{S_{1}, S_{2}, \ldots, S_{L} \}$ on $D\subseteq \boldsymbol{\mathbbm{R} } ^{\! N}$ then, there is a unique attractor $F\subseteq D$ satisfying the identity \eqref{Def Attractor of an IFS}.
\end{theorem}
\begin{proof}
See \textit{Theorem} 9.1 \cite{Falconer}.
\qed
\end{proof}
\begin{remark}\label{Rem The Attractor}
In this work, the attractor $F$ of an IFS will prove to be important in an indirect way: not for the definition of a microstructure, but for analyzing the nature of the attained conclusions.
\end{remark}
It is a well-known fact that under certain conditions (see \cite{Falconer}, \textit{Lemma} 9.2) a strictly self-similar set $F$ has both, Hausdorff and Box dimensions which are equal (see \cite{Falconer}, \textit{Theorem} 9.3), namely $d$. Moreover, if $c_{i}$ is the ratio of the similarity $S_{i}$, then
\begin{equation}\label{Eq Dimension of Strictly Self-Similar Sets}
\sum\limits_{i\, =\, 1}^{L} c_{i}^{d} = 1\, .
\end{equation}
From now on, we limit our attention to fractal structures satisfying strict self-similarity.
Finally, we introduce the type of microstructure to be studied in throughout this work.
\begin{definition}[Fractal Microstructure]\label{Def Fractal Microstructure}
We say that a set $\mathsf{B} \subseteq [0,1]$ is a \textbf{fractal microstructure} if it is countable and there exists a sequence of finite subsets $\{\mathsf{B} _{n}: n \geq 0\}$, together with an iterated function system of similarities $\{S_{i}: 1\leq i\leq L \}$ on $[0,1]$, satisfying the following conditions
\begin{enumerate}[(i)]

\item The set $\mathsf{B} _{0}$ is finite and $\mathsf{B} _{n}$ is recursively defined by
\begin{align}\label{Eq Fractal Microstructure}
& \mathsf{B} _{n} = \bigcup\limits_{i\,=\, 1}^{L} S_{i}(\mathsf{B} _{n - 1})\, ,  &
& \text{for all }\; n\in \boldsymbol{\mathbbm{N} } \, .
\end{align}

\item The sequence of sets is monotonically increasing and
\begin{equation}\label{Eq sigma development}
\mathsf{B}_{n} \underset{n}\uparrow \mathsf{B} \, .
\end{equation}

\end{enumerate}
In the following, we refer to $\{\mathsf{B} _{n}: n\geq 0\}$ as a $\bm{\sigma}$\textbf{-finite development} of $\mathsf{B} $.
\end{definition}
\begin{remark}\label{Rem Properties of the sigma finite development}
Let $\mathsf{B} \subseteq [0,1]$ be a fractal microstructure and let $\{S_{i}: 1\leq i\leq L \}$ be its corresponding system of similarities, notice the following
\begin{enumerate}[(i)]
\item There may exist more than one $\sigma$-finite development of $\mathsf{B} $.

\item Given a $\sigma$-finite development $\{\mathsf{B} _{n}: n \geq 0\}$ then, for each $n \in \boldsymbol{\mathbbm{N} } $, the following relationships of cardinality must hold
\begin{align}\label{Eq Cardinality Relationships}
& \card(\mathsf{B} _{n})  \leq  L\, \card(\mathsf{B} _{n-1}) \, , &
& \card(\mathsf{B} _{n})  \leq  L^{n}\, \card(\mathsf{B} _{0}) \,   , &
& \card(\mathsf{B} _{n} - \mathsf{B} _{n -1}) \leq  L^{n -1}(L -1) \, \card(\mathsf{B} _{0})  \,  .
\end{align}

\item The fractal microstructure $\mathsf{B} $ is necessarily contained in the unique fractal attractor $F$ of the IFS of similarities $\{S_{i}: 1\leq i\leq L \}$.
\end{enumerate}
\end{remark}
\section{Unscaled Storage Model for the Interface Microstructure}\label{Sec A sigma finite Interface Microstructure}
Let $\mathsf{B} $ be a microstructure set, let $\{\mathsf{B}_{n} :n\geq 0 \}$ be a $\sigma$-finite development and consider the following sequence of strong interface problems
\begin{subequations}\label{Pblm strong problem n-stage}
   \begin{equation}\label{Eq strong n-stage equation}
      - \boldsymbol{\partial } ^{2} p_{n} = F \quad \text{in}\; \left[\mathpzc{b} _{k-1}, \mathpzc{b} _{k}\right] .
    \end{equation}
    %
    With the interface conditions
    %
    \begin{equation}\label{Eq strong n-stage interface}
    %
    \begin{split}
    p_{n}  \left(\mathpzc{b} _{k}^{-}\right) & = p_{n} \left(\mathpzc{b} _{k}^{+}\right)  ,
    \\

    \boldsymbol{\partial }  p_{n}  \left(\mathpzc{b} _{k}^{-}\right)  -  \boldsymbol{\partial }  p_{n}  \left(\mathpzc{b} _{k}^{+}\right) +
   \beta\,  p_{n}  \left(\mathpzc{b} _{k}\right) & =  f\left(\mathpzc{b} _{k}\right) \, ,
    \quad \forall \; 1\leq k\leq K_{n} - 1 \, .
    \end{split}

    \end{equation}
    %
    And the boundary conditions
    %
    \begin{align}\label{Eq strong n-stage boundary}
      & p_{n}  (0)= 0 \, , &
      & \boldsymbol{\partial }  p_{n} (1)= 0 \,.
    \end{align}

\end{subequations}
Here, $0 = \mathpzc{b} _{0} < \mathpzc{b} _{1} < \mathpzc{b} _{2}< \ldots < \mathpzc{b} _{\scriptscriptstyle K_{n}} = 1$ is a monotone ordering of $\mathsf{B}_{n} $ i.e., $K_{n} = \card(\mathsf{B}_{n} )$. The forcing term $F$ belongs to $L^{2}(0,1)$, $\beta > 0$ is a storage fluid exchange coefficient, and $f$ is a source on the interface. In order to attain the variational formulation of the problems above we define the following function space
\begin{definition}\label{Def function space setting}
Define the space
\begin{equation}\label{Def space of pressures direct}
V \defining \big\{u\in H^{\,1}(0,1): u(0) = 0 \big\} ,
\end{equation}
endowed with the inner product $\langle \cdot, \cdot\rangle_{\scriptscriptstyle V}: V\times V\rightarrow \boldsymbol{\mathbbm{R} } $
\begin{equation}\label{Def norm space of pressures direct}
\langle u, v\rangle_{\scriptscriptstyle V}\defining \int_{0}^{1} \boldsymbol{\partial }  u \, \boldsymbol{\partial }  v \, ,
\end{equation}
and the norm $\Vert u\Vert_{\scriptscriptstyle V} \defining \sqrt{\langle u, u\rangle_{\scriptscriptstyle V}} $.
\end{definition}
The variational formulation of the \textit{Problem} \eqref{Pblm strong problem n-stage} above is given by
\begin{align}\label{Pblm Variational n stage Concise}
& p_{n}  \in V : &
& \int_{0}^{1}\boldsymbol{\partial }  p_{n}  \, \boldsymbol{\partial }  q
+ \beta \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n}  } p_{n}   (\mathpzc{b} ) \,  q(\mathpzc{b} )
= \int_{0}^{1}F\,q
+ \sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} } f(\mathpzc{b} ) \,  q(\mathpzc{b} )\, , &
& \forall \, q\in V.
\end{align}
\begin{theorem}\label{Th boundedness of the sequence of solutions}
The Problem \eqref{Pblm Variational n stage Concise} is well-posed. Moreover if $f\in \ell^{2}(\mathsf{B} )$ the sequence of solutions $\{p_{n} : n\in \boldsymbol{\mathbbm{N} } \}$ is bounded in $V$ and for each $n\in \boldsymbol{\mathbbm{N} } $ it holds that
\begin{equation}\label{Ineq a-priori estimate pressures}
\begin{split}
\Vert p_{n} \Vert_{V},\, \Big(\sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} } p_{n} ^{2}(\mathpzc{b} )\Big)^{ 1/2 }
\leq
\frac{1}{\min\{1, \beta\} }
\Big(\Vert F \Vert_{L^{2} (0,1)}^{2} +\Vert f\Vert^{2}_{\ell^{2}(\mathsf{B} )}\Big)^{ 1/2 } .
\end{split}
\end{equation}
\end{theorem}
\begin{proof}
Clearly, the bilinear form $(q, r)\mapsto \int_{0}^{1}\boldsymbol{\partial }  q \, \boldsymbol{\partial }  r
+ \beta \sum\limits_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n}  } q (\mathpzc{b} ) \,  r(\mathpzc{b} )$ is continuous and $V$-elliptic. In addition, it is direct to see that $q\mapsto \int_{0}^{1}F\,q + \sum\limits_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} } f(\mathpzc{b} ) \,  q(\mathpzc{b} )$ is linear and continuous. Consequently, the well-posedness of \textit{Problem} \eqref{Pblm Variational n stage Concise} follows from the Lax-Milgram Theorem (see  \cite{Showalter77}). For the boundedness of $\{p_{n} : n\in \boldsymbol{\mathbbm{N} }  \}$, we test \eqref{Pblm Variational n stage Concise} with $p_{n} $ and get
\begin{equation} \label{Ineq Cauchy - Schwartz in n-stage problem}
\begin{split}
\int_{0}^{1}(\boldsymbol{\partial }  p_{n} )^{2}
+ \beta \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n}  }  p_{n} ^{2}  (\mathpzc{b} ) &
= \int_{0}^{1}F\,p_{n}
+ \sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} } f(\mathpzc{b} ) \,  p_{n} (\mathpzc{b} )  \\
& \leq \Vert F \Vert_{L^{2} (0,1)} \Vert p_{n}  \Vert_{L^{2} (0,1)}
+ \bigg(\sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} } f^{2}(\mathpzc{b} )\bigg)^{1/2} \,
\bigg(\sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} } p_{n} ^{2}(\mathpzc{b} ) \bigg)^{1/2}\\
& \leq
\bigg(\Vert F \Vert_{L^{2} (0,1)}^{2} + \sum_{\mathpzc{b}  \, \in \, \mathsf{B} } f^{2}(\mathpzc{b} )\bigg)^{ 1/2 }
\bigg(\K^{2}_{(0,1)} \, \Vert \boldsymbol{\partial }  p_{n}  \Vert_{L^{2} (0,1)}^{2} +
\sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} } p_{n} ^{2}(\mathpzc{b} )\bigg)^{ 1/2 }.
\end{split}
\end{equation}
The second and third lines were obtained applying the Cauchy-Schwartz inequality in $L^{2}(0,1)$, $\boldsymbol{\mathbbm{R} } ^{\card(\mathsf{B}_{n} ) }$ and $\boldsymbol{\mathbbm{R} } ^{2}$ respectively. Also, $\K_{(0,1)}$ is the Poincar\'e constant associated to the domain $(0,1)$; in this particular case $\K_{(0,1)}\leq \dfrac{1}{\sqrt{2}}$  (see \cite{Showalter77}). Hence,
\begin{equation*}
\begin{split}
\max \Big\{\Vert p_{n} \Vert_{V},
\bigg(\sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} } p_{n} ^{2}(\mathpzc{b} )\bigg)^{ 1/2 } \Big\}
& \leq
\Big(\int_{0}^{1}(\boldsymbol{\partial }  p_{n} )^{2}  + \sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} } p_{n} ^{2}(\mathpzc{b} )\Big)^{  1/2 } \\
& \leq \frac{1}{\min\{1, \beta\} }
\Big(\Vert F \Vert_{L^{2} (0,1)}^{2} +\Vert f\Vert^{2}_{\ell^{2}(\mathsf{B} )}\Big)^{ 1/2 }
\quad \forall\, n\in \boldsymbol{\mathbbm{N} }  .
\end{split}
\end{equation*}
Hence, the \textit{Estimate} \eqref{Ineq a-priori estimate pressures} follows.
\qed
\end{proof}
In the theorem above, particularly due to the a-priori \textit{Estimate} \eqref{Ineq a-priori estimate pressures}, we observe that the sequence of solutions $\{p_{n}: n\in \boldsymbol{\mathbbm{N} } \}\subseteq V$ contains extra information, which is given by the boundedness of the term $\sum\limits_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} } p_{n} ^{2}(\mathpzc{b} )$. Implicitly, this fact gives the subspace of convergence.
\begin{definition}\label{Def Space of Convergence}
Define the \textbf{Fractal Interface Space}
\begin{equation}\label{Def fractal space pressures direct}
V_{ \B} \defining \Big\{u\in H^{\,1}(0,1): u(0) = 0,
\sum_{\mathpzc{b} \,\in\, \mathsf{B} }\vert u(\mathpzc{b} ) \vert^{2} <\infty \Big\} .
\end{equation}
Endowed with the inner product $\langle \cdot, \cdot\rangle: V_{ \B}\timesV_{ \B}\rightarrow \boldsymbol{\mathbbm{R} } $
\begin{equation}\label{Def norm fractal space of pressures direct}
\langle u, v\rangle_{\scriptscriptstyle V_{ \B}}\defining \int_{0}^{1} \boldsymbol{\partial }  u \, \boldsymbol{\partial }  v
+ \sum_{\mathpzc{b} \,\in\, \mathsf{B} }u(\mathpzc{b} )\, v(\mathpzc{b} ),
\end{equation}
and the norm $\Vert u\Vert_{\scriptscriptstyle V_{ \B}}^{2} \defining \langle u, u\rangle_{\scriptscriptstyle V_{ \B}} $.
\end{definition}
\begin{remark}\label{Rem the little ell 2 equivalence}
\begin{enumerate}[(i)]
\item In the following we refer to the trace operator $\gamma_{\scriptscriptstyle \mathsf{B} }: V_{ \B} \rightarrow \boldsymbol{\mathbbm{R} } ^{\!\mathsf{B} }$, $u\mapsto u\vert_{\mathsf{B} }$ as the \textbf{fractal trace operator} on $\mathsf{B} $, which is well-defined for functions $u\in V$.

\item
The space $V_{ \B}$ is the subspace of functions $u \in V$ whose trace on the microstructure set, $u\vert_{\mathsf{B} }$, belongs to $\ell^{\, 2}(\mathsf{B} )$.

\item
The space $V_{ \B}$ is a Hilbert space due to the completeness of $V$ with the norm $\Vert \cdot \Vert_{\scriptscriptstyle V} $ and the completeness of $\ell^{\, 2}(\mathsf{B} )$ with the norm $\big\{\sum\limits_{\mathpzc{b} \,  \in \, \mathsf{B} } u^{2}(\mathpzc{b} )\big\}^{ 1/2 }$.

\end{enumerate}

\end{remark}
\begin{theorem}\label{Th weak convergence of the whole sequence}
Let $\{p_{n} : n\in \boldsymbol{\mathbbm{N} } \}\subseteq V$ be the sequence of solutions to the family of \textit{Problems} \eqref{Pblm Variational n stage Concise}. Then, if it converges weakly in $V$ to an element $\xi\in V$, the sequence of functionals
$\Lambda_{n}:\ell^{\, 2} (\mathsf{B} )\rightarrow \boldsymbol{\mathbbm{R} } $ defined by
\begin{equation}\label{Def partial development functionals}
\Lambda_{n}(u) \defining \sum_{\mathpzc{b}  \,\in \, \mathsf{B}_{n} } p_{n}(\mathpzc{b} )\, u(\mathpzc{b} )
= \sum_{\mathpzc{b}  \,\in \, \mathsf{B} } p_{n}(\mathpzc{b} )\, u(\mathpzc{b} ) \, \boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n} }(\mathpzc{b} )\,,\quad \forall \,  u\in \ell^{\, 2}(\mathsf{B} ),
\end{equation}
converges weakly to $\xi\vert_{\mathsf{B} }$ in $(\ell^{2}(\mathsf{B} ))'$. In particular, $\xi\in V_{ \B}$ and
\begin{equation}\label{Eq convergence of the fractal functional}
\sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} }  p_{n} (\mathpzc{b} ) \, q(\mathpzc{b} ) \xrightarrow[n\rightarrow \infty]{}
\sum_{\mathpzc{b}  \, \in \, \mathsf{B} }  \xi(\mathpzc{b} ) \, q(\mathpzc{b} ) \, , \quad
\forall\, q\in V_{ \B}.
\end{equation}
\end{theorem}
\begin{proof}
Clearly $\Lambda_{n}\in \ell^{\, 2} (\mathsf{B} ) '$ for all $n\in \boldsymbol{\mathbbm{N} } $ and due to the Riesz Representation theorem, we have that
\begin{equation*}
\Vert\Lambda_{n}\Vert = \Big(\sum_{\mathpzc{b}  \,\in \, \mathsf{B}_{n} } p_{n}^{2}(\mathpzc{b} ) \Big)^{1/2}
\leq \frac{M}{\min \{ 1, \beta \} }\quad \forall\, n\in \boldsymbol{\mathbbm{N} }  ,
\end{equation*}
where the last inequality holds due to the \textit{Estimate} \eqref{Ineq a-priori estimate pressures}. Consequently, there must exist a subsequence $\{n_{k}: k\in \boldsymbol{\mathbbm{N} }  \}$ and an element $\eta\in \ell^{\, 2}(\mathsf{B} )$ such that
\begin{equation*}
\Lambda_{n_{k} }(u) =  \sum_{\mathpzc{b}  \,\in \, \mathsf{B}_{n_{k} } } p_{n_{k}} (\mathpzc{b} )\, u(\mathpzc{b} ) \xrightarrow[k \, \rightarrow \, \infty]{}
\sum_{\mathpzc{b}  \, \in \, \mathsf{B} } \eta (\mathpzc{b} )\, u(\mathpzc{b} )
\,,\quad \forall \,  u\in \ell^{ \, 2}(\mathsf{B} ).
\end{equation*}
In particular, for any $\mathpzc{b} _{0} \in \mathsf{B} $, let $K\in \boldsymbol{\mathbbm{N} } $ be such that $\mathpzc{b} _{0}\in \mathsf{B}_{n_{k} } $ for all $k>K$ then, recalling that $\boldsymbol{\mathbbm{1} } _{ \{\mathpzc{b} _{0}\} } \in \ell^{\,2}(\mathsf{B} )$, the expression above yields
\begin{align*}
& p_{n_{k}} (\mathpzc{b} _{0}) =  \sum_{\mathpzc{b}  \,\in \, \mathsf{B}_{n_{k} } } p_{n_{k}} (\mathpzc{b} )\, \boldsymbol{\mathbbm{1} } _{\{\mathpzc{b} _{0}\}}(\mathpzc{b} )  \xrightarrow[k \, > \, K]{}
\sum_{\mathpzc{b}  \, \in \, \mathsf{B} } \eta (\mathpzc{b} )\, \, \boldsymbol{\mathbbm{1} } _{\{\mathpzc{b} _{0}\}}(\mathpzc{b} ) = \eta (\mathpzc{b} _{0})
\,, &
& \forall \; \mathpzc{b} _{0}\in\mathsf{B} .
\end{align*}
On the other hand, since $p_{n} \overset{w}\rightharpoonup \xi$ this implies that $p_{n} (x_{0})\rightarrow \xi(x_{0})$ for all $x_{0} \in [0,1]$.  In particular $\lim\limits_{\, k \, \rightarrow \, \infty}p_{n_{k}} (\mathpzc{b} _{0})  = \xi(\mathpzc{b} _{0})$ for all $\mathpzc{b} _{0} \in \mathsf{B} $, consequently $\xi\vert_{\mathsf{B} } = \eta$ and  $\xi\in V_{ \B}$. Moreover, since the above holds for any convergent subsequence of $\{\Lambda_{n}: n\in \boldsymbol{\mathbbm{N} } \}\subseteq (\ell^{\, 2}(\mathsf{B} ))'$ it follows that the whole sequence is weakly convergent i.e.,
\begin{equation*}
\Lambda_{n} \overset{w}\rightharpoonup \xi\vert_{\mathsf{B} }\, ,  \text{weakly in}\; (\ell^{\, 2}(\mathsf{B} ))' .
\end{equation*}
From here, the convergence statement \eqref{Eq convergence of the fractal functional} follows trivially.
\qed
\end{proof}
\begin{remark}\label{Rem partial interface development functionals}
It is important to observe that the weak convergence of the functionals $\Lambda_{n}$ in $(\ell^{\, 2}(\mathsf{B} ))'$ hypothesis is a stronger condition than the \textit{Statement} \eqref{Eq convergence of the fractal functional}, since it can not be claimed that the fractal trace operator $\gamma_{\scriptscriptstyle \mathsf{B} }: V_{ \B} \rightarrow \ell^{\, 2}(\mathsf{B} )$, $q\mapsto q\vert_{\mathsf{B} }$ is surjective. Moreover, it will be shown that in most of the interesting cases this operator is not surjective.
\end{remark}
\subsection{The Limit Problem}\label{Sec the limit problem}
Consider the variational problem with microstructure interface
\begin{align}\label{Pblm Variational Microstructure Concise}
& p \in V_{ \B} : &
& \int_{0}^{1}\boldsymbol{\partial }  p \, \boldsymbol{\partial }  q
+ \beta \sum_{\mathpzc{b}  \, \in  \, \mathsf{B} } p  (\mathpzc{b} ) \,  q(\mathpzc{b} )= \int_{0}^{1}F\,q
+ \sum_{\mathpzc{b}  \, \in \, \mathsf{B} } f(\mathpzc{b} ) \,  q(\mathpzc{b} )\,, &
& \forall \, q\in V_{ \B}.
\end{align}
Where, $F\in L^{2}(0,1)$ and $f\in \ell^{\, 2}(\mathsf{B} )$. We claim that the solution $p$ of the problem above is the weak limit $\{p_{n} : n\in \boldsymbol{\mathbbm{N} }  \}\subseteq V$ i.e., \textit{Problem} \eqref{Pblm Variational Microstructure Concise} is the ``\textbf{limit}" of \textit{Problems} \eqref{Pblm Variational n stage Concise}.
\begin{theorem}\label{Th well-posedness fractal interface problem}
The \textit{Problem} \eqref{Pblm Variational Microstructure Concise} is well-posed.
\end{theorem}
\begin{proof}
Consider the bilinear form
\begin{equation}\label{Def Bilinear Form Dyadic}
a(q, r)\defining \int_{0}^{1}\boldsymbol{\partial }  q \, \boldsymbol{\partial }  r
+ \beta \sum_{\mathpzc{b}  \, \in  \, \mathsf{B} } q  (\mathpzc{b} ) \,  r(\mathpzc{b} ).
\end{equation}
Using the Cauchy-Schwartz inequality in each summand of the expression above, we conclude the continuity of the bilinear form.
On the other hand, it is direct to see that $\min\{1, \beta\}\Vert q \Vert_{\scriptscriptstyle V_{ \B}}^{2}\leq \vert a(q,q)\vert$, which implies that the bilinear form is $V_{ \B}$-elliptic. Applying the Lax-Milgram Theorem, the result follows, see \cite{Showalter77}.
\qed
\end{proof}
Now we are ready to prove the weak convergence of the whole sequence of solutions of \textit{Problems} \eqref{Pblm Variational n stage Concise} to the solution $p$ of Problem \eqref{Pblm Variational Microstructure Concise}.
\begin{theorem}\label{Th weak convergence of the solutions in V}
Let $\{p_{n} : n\in \boldsymbol{\mathbbm{N} } \}\subseteq V$ be the sequence of solutions of \textit{Problems} \eqref{Pblm Variational n stage Concise}, then, it converges weakly in $V$ to the unique solution $p$ of \textit{Problem} \eqref{Pblm Variational Microstructure Concise}.
\end{theorem}
\begin{proof}
Since $\{p_{n} : n\in \boldsymbol{\mathbbm{N} } \}$ is bounded in $V$ there must exist a weakly convergent subsequence $\{p_{n_{k}} : k\in \boldsymbol{\mathbbm{N} }  \}$ and a limit $\xi\in V$. Test \eqref{Pblm Variational n stage Concise} with $q\in V_{ \B}\subseteq V$ arbitrary, this gives
\begin{equation*}
\int_{0}^{1}\boldsymbol{\partial }  p_{n_{k}}   \boldsymbol{\partial }  q  +
 \beta \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n_{k} }  } p_{n_{k}}   (\mathpzc{b} ) \,  q(\mathpzc{b} )= \int_{0}^{1}F\,q
+ \sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{n_{k} } } f(\mathpzc{b} ) \,  q(\mathpzc{b} ) \, .
\end{equation*}
Letting $k\rightarrow \infty$ in the expression above and, in view of \textit{Theorem} \ref{Th weak convergence of the whole sequence}, we get
\begin{align*}
& \int_{0}^{1}\boldsymbol{\partial }  \xi \, \boldsymbol{\partial }  q  +
 \beta \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}  } \xi  (\mathpzc{b} ) \,  q(\mathpzc{b} ) = \int_{0}^{1}F\,q
+ \sum_{\mathpzc{b}  \, \in \, \mathsf{B} } f(\mathpzc{b} ) \,  q(\mathpzc{b} ) \,, &
& \forall \, q\in V_{ \B} .
\end{align*}
Additionally, \textit{Theorem} \ref{Th weak convergence of the whole sequence} implies that $\xi\in V_{ \B}$. Therefore $\xi$ is a solution to \textit{Problem} \eqref{Pblm Variational Microstructure Concise}, which is unique due to \textit{Theorem} \ref{Th well-posedness fractal interface problem}; therefore we conclude that $\xi = p$. Since the above holds for any weakly convergent subsequence of $\{p_{n} : n \in \boldsymbol{\mathbbm{N} }  \}$ it follows that the whole sequence must converge weakly to $p$.
\qed
\end{proof}
\begin{lemma}\label{Th norms convergence}
Let $\{p_{n} : n\in \boldsymbol{\mathbbm{N} }  \}\subseteq V$ be the sequence of solutions of \textit{Problems} \eqref{Pblm Variational n stage Concise} then
\begin{subequations}\label{Stmt strong convergence}
\begin{equation} \label{Stmt strong convergence in V}
\Vert p_{n} -  p \Vert_{\scriptscriptstyle V} \xrightarrow[ n \, \rightarrow \, \infty ]{} 0 .
\end{equation}
\begin{equation} \label{Stmt strong convergence of interface fractal norm}
\left\Vert \gamma_{\scriptscriptstyle \mathsf{B} } (p_{n} ) \boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n} } -  \gamma_{\scriptscriptstyle \mathsf{B} }(p)\,\boldsymbol{\mathbbm{1} } _{\mathsf{B} }\right\Vert_{\scriptscriptstyle \ell^{\, 2}(\mathsf{B} )} \xrightarrow[n \, \rightarrow \, \infty]{} 0 .
\end{equation}
Where $\gamma_{\scriptscriptstyle \mathsf{B} }(q) \defining q\,\boldsymbol{\mathbbm{1} } _{\scriptscriptstyle \mathsf{B}  }$ is the fractal trace operator on $V$.
\end{subequations}
\end{lemma}
\begin{proof}
We know that $p_{n}  \overset{w}\rightharpoonup p$ weakly in $V$ and $p_{n}  \boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n} }\overset{w}\rightharpoonup p\boldsymbol{\mathbbm{1} } _{\mathsf{B} }$ weakly in $\ell^{\,2}(\mathsf{B} )$ from \textit{Theorems} \ref{Th weak convergence of the solutions in V} and \ref{Th weak convergence of the whole sequence} respectively, therefore
\begin{align*}
\int_{0}^{1}\left\vert \boldsymbol{\partial }  p\right\vert^{2}  \leq
\liminf_{n}\int_{0}^{1} \left\vert \partial p_{n} \right\vert^{2} ,\\
\sum_{\mathpzc{b} \,\in\, \mathsf{B} }p^{2}(\mathpzc{b} ) \leq  \liminf_{n} \sum_{\mathpzc{b} \,\in\, \mathsf{B} }p_{n} ^{2}(\mathpzc{b} )\,\boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n} }(\mathpzc{b} ) =
\liminf_{n} \sum_{\mathpzc{b} \,\in\, \mathsf{B}_{n} }p_{n} ^{2}(\mathpzc{b} ) .
\end{align*}
On the other hand, testing \eqref{Pblm Variational Microstructure Concise} on the diagonal $p_{n} $, we get
\begin{equation*}
\int_{0}^{1}\boldsymbol{\partial }  p_{n} ^{2}
+ \beta \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}  } p_{n} ^{2}  (\mathpzc{b} )\,\boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n} }(\mathpzc{b} ) = \int_{0}^{1}F\,p_{n}
+ \sum_{\mathpzc{b}  \, \in \, \mathsf{B} } f(\mathpzc{b} ) \,  p_{n} (\mathpzc{b} ) \, \boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n} }(\mathpzc{b} ) .
\end{equation*}
Letting $n\rightarrow\infty$ in the expression above we get
\begin{equation}\label{Ineq Solution Sandwich}
\begin{split}
\lim_{n}\Big\{\int_{0}^{1}\boldsymbol{\partial }  p_{n} ^{2}
+ \beta \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n}  } p_{n} ^{2}  (\mathpzc{b} )\Big\}
& = \int_{0}^{1}F\,p
+ \sum_{\mathpzc{b}  \, \in \, \mathsf{B} } f(\mathpzc{b} ) \,  p(\mathpzc{b} )  \\
& = \int_{0}^{1}\boldsymbol{\partial }  p^{2}
+ \beta \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}  } p^{2}  (\mathpzc{b} ) \\
& \leq \liminf_{n}\int_{0}^{1} \left\vert \partial p_{n} \right\vert^{2}
+  \beta\, \liminf_{n}  \sum_{\mathpzc{b} \,\in\, \mathsf{B}_{n} }p_{n} ^{2}(\mathpzc{b} ) .
\end{split}
\end{equation}
Hence
\begin{equation*}
\lim_{n}\Big\{\int_{0}^{1}\boldsymbol{\partial }  p_{n} ^{2}
+ \beta \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n}  } p_{n} ^{2}  (\mathpzc{b} ) \Big\}
= \liminf_{n}\int_{0}^{1} \left\vert \partial p_{n} \right\vert^{2}
+  \beta\, \liminf_{n}  \sum_{\mathpzc{b} \,\in\, \mathsf{B}_{n} }p_{n} ^{2}(\mathpzc{b} ) .
\end{equation*}
From here, a simple exercise of real sequences shows that both sequences $\{\int_{0}^{1} \left\vert \partial p_{n} \right\vert^{2}: n\in \boldsymbol{\mathbbm{N} }  \}$ and $\big\{\sum\limits_{\mathpzc{b} \,\in\, \mathsf{B}_{n} }p_{n} ^{2}(\mathpzc{b} ): n\in \boldsymbol{\mathbbm{N} } \big\}$ converge.
Combining these facts with \textit{Inequality} \eqref{Ineq Solution Sandwich} we have
\begin{equation*}
\int_{0}^{1}\boldsymbol{\partial }  p^{2}
+ \beta \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}  } p^{2}  (\mathpzc{b} ) \\
= \lim_{n}\int_{0}^{1}\boldsymbol{\partial }  p_{n} ^{2}
+ \beta \, \lim_{n} \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n}  } p_{n} ^{2}  (\mathpzc{b} ).
\end{equation*}
Therefore, if $\int_{0}^{1}\boldsymbol{\partial }  p^{2} \lneqq \liminf\limits_{n}\int_{0}^{1} \left\vert \partial p_{n} \right\vert^{2}$ or $\sum\limits_{\mathpzc{b}  \, \in  \, \mathsf{B}  } p^{2}  (\mathpzc{b} )\lneqq \liminf\limits_{n} \sum\limits_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n}  } p_{n} ^{2}  (\mathpzc{b} )$, the equality above could not be possible. Then, it holds that
\begin{equation*}
\Vert p \Vert_{\scriptscriptstyle V}^{2}
= \lim_{n}\int_{0}^{1}  \partial p_{n} ^{2}
= \lim_{n} \Vert p_{n}  \Vert_{\scriptscriptstyle V}^{2} ,
\end{equation*}
and
\begin{equation*}
\Vert \gamma_{\scriptscriptstyle \mathsf{B} }(p) \Vert_{\scriptscriptstyle \ell^{2}(\mathsf{B} )}^{2}
= \lim_{n} \sum_{\mathpzc{b} \,\in\, \mathsf{B} }p_{n} ^{2}(\mathpzc{b} )  \, \boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n} }(\mathpzc{b} )
=
\lim_{n} \Vert \gamma_{\scriptscriptstyle \mathsf{B} }(p_{n} \boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n} }) \Vert_{\scriptscriptstyle \ell^{2}(\mathsf{B} )}^{2}.
\end{equation*}
Finally, the convergence of norms together with the weak convergence on the underlying spaces, imply the strong convergence \eqref{Stmt strong convergence} of both sequences.
\qed
\end{proof}
\subsection{The Space $V_{ \B}$}
We start this section proving a lemma which is central in the understanding of the space $V_{ \B}$.
\begin{lemma}\label{Th microstructure accumulation points}
Let $\mathsf{B} $ be a microstructure in $[0,1]$ and let $q\in H^{1}(0,1)$ be such that $q\vert_{\mathsf{B} }\in \ell^{p}(\mathsf{B} )$ with $1\leq p < \infty$. Then, if $\mathpzc{b} _{0}\in \mathsf{B} $ is an accumulation point of $\mathsf{B} $, it must hold that $q(\mathpzc{b} _{0}) = 0$.
\end{lemma}
\begin{proof}
Let $q$ satisfy the hypotheses and let $\mathpzc{b} _{0}\in \mathsf{B} $ be an accumulation point of $\mathsf{B} $ such that $\vert q(\mathpzc{b} _{0}) \vert >0$. Since $q\in H^{1}(0,1)$, it is absolutely continuous and there exists $\epsilon >0$ such that $\vert q(x)\vert > \dfrac{1}{2}\, \vert q(\mathpzc{b} _{0}) \vert$ for all $\vert x - \mathpzc{b} _{0}\vert < \epsilon$; hence
\begin{equation*}
\sum_{\mathpzc{b}  \, \in \, \mathsf{B} } \vert q(\mathpzc{b} ) \vert^{p} \geq
\sum_{\substack{\mathpzc{b} \, \in \,\mathsf{B} \\[3pt] \vert \mathpzc{b}  - \mathpzc{b} _{0}\vert < \epsilon }} \vert q(\mathpzc{b} ) \vert^{p}
\geq
 \card \big(\{ \mathpzc{b} \, \in \,\mathsf{B}  : \ \vert \mathpzc{b}  - \mathpzc{b} _{0}\vert < \epsilon\}\big) \, \frac{1}{2^{p}}\,\vert q(\mathpzc{b} _{0}) \vert^{p}.
\end{equation*}
However, the set $\{ \mathpzc{b} \, \in \,\mathsf{B}  : \ \vert \mathpzc{b}  - \mathpzc{b} _{0}\vert < \epsilon\}$ contains infinitely many points because $\mathpzc{b} _{0}$ is an accumulation point of $\mathsf{B} $, therefore $q\vert_{\mathsf{B} }$ does not belong to $\ell^{p}(\mathsf{B} )$ which is absurd.
\qed
\end{proof}
\begin{corollary}\label{Th dense microstructure}
Suppose that $\mathsf{B} $ is dense in $[0,1]$ then $V_{ \B} = \{0\}$.
\end{corollary}
\begin{proof}
Due to \textit{Lemma} \ref{Th microstructure accumulation points} if $q\in V_{ \B}$ it must hold that $q(\mathpzc{b} ) = 0$ for every $\mathpzc{b} $ accumulation point of $\mathsf{B} $. Since $\mathsf{B} $ is dense in $(0,1)$ every point of $\mathsf{B} $ is an accumulation point of $\mathsf{B} $, therefore $q(\mathpzc{b} ) = 0$ for all $\mathpzc{b} \in \mathsf{B} $. On the other hand, $q$ is absolutely continuous because it belongs to $H^{1}(0,1)$ and due to the density of $\mathsf{B} $ in $[0,1]$ it follows that $q = 0$.
\qed
\end{proof}
The \textit{Lemma} \ref{Th microstructure accumulation points} states that the accumulation points of the microstructure contained in $\mathsf{B} $ define an important property of $V_{ \B}$. In addition, the next property follows trivially
\begin{corollary}\label{Th dense microstructure trivializes}
Let $\mathsf{B} \subseteq [0,1]$ be a dense microstructure then, the \textit{Problem} \eqref{Pblm Variational Microstructure Concise} becomes trivial and the sequence $\{p_{n} : n\in \boldsymbol{\mathbbm{N} }  \}$ satisfies
\begin{align}\label{Stmt convergence to zero dense microstructure}
& \Vert p_{n} \Vert_{\scriptscriptstyle V} \rightarrow 0 \; , &
& \sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} }p_{n} ^{2}(\mathpzc{b} ) \rightarrow 0 \; .
\end{align}
\end{corollary}
The facts presented in \textit{Lemma} \ref{Th microstructure accumulation points} and \textit{Corollary} \ref{Th dense microstructure trivializes} are unfortunate, since several important fractal microstructures are dense in $[0,1]$. On the other hand, if a microstructure $\mathsf{B} $ is not dense in $[0,1]$ but it is a perfect set (which is also an important case) the \textit{Problem} \eqref{Pblm Variational Microstructure Concise}, without becoming trivial, becomes fully decoupled and consequently uninteresting. An important example of the first case are the Dyadic numbers in $[0,1]$ and an example of the second case is the collection of extremes of the removed intervals in the construction of the Cantor set.

As an alternative, it is possible to strengthen the conditions on the interface forcing term $f$, seeking to weaken the summability properties of the limit function fractal trace $\gamma_{\scriptscriptstyle \mathsf{B} }(p) = p\boldsymbol{\mathbbm{1} } _{\mathsf{B} }$. However, this approach yields estimates equivalent to \textit{Inequality} \eqref{Ineq a-priori estimate pressures} in \textit{Theorem} \ref{Th boundedness of the sequence of solutions} and consequently $p\in V_{ \B}$. Therefore, this is not a suitable choice either.

\begin{comment}
\subsection{Two Particular Microstructures}\label{Sec Two particular microstructures}
The facts presented in \textit{Lemma} \ref{Th microstructure accumulation points} and Corollary \ref{Th dense microstructure trivializes} are unfortunate, since several important fractal microstructures are dense in $[0,1]$ or perfect sets. On the other hand, if a microstructure $\mathsf{B} $ is not dense in $[0,1]$ but it is a perfect set, the \textit{Problem} \eqref{Pblm Variational Microstructure Concise} without becoming trivia, becomes nonetheless uninteresting. In this section we examine to particular important examples of microstructures.
\begin{example}\label{Def Dyadic Microstructure}
The \textbf{dyadic numbers} are given by
\begin{subequations}\label{Def Dyadic Numbers}
\begin{equation}\label{Def Dyadic Set}
\mathcal{D}  \defining \bigcup_{n \, \in \, \boldsymbol{\mathbbm{N} } }
\bigcup_{k \, = \, 0}^{ 2^{n} }
\Big\{\frac{k}{2^{n}}\Big\} .
\end{equation}
Consider the following $n$-stage partial development
\begin{align}\label{Def Partial Dyadic Set}
&\mathcal{D}_{n}  \defining \bigcup_{k = 0}^{ 2^{n} } \Big\{\frac{k}{2^{n}}\Big\} ,&
& \mathcal{I}_{n}  \defining \bigcup_{k = 1}^{ 2^{n} - 1} \Big\{\frac{k}{2^{n}}\Big\} .
\end{align}
With $\mathcal{I}_{n} $ the interior points of the $n$-stage development of the interface microstructure set $\mathcal{D}  - \{ 0, 1\}$. The indicator function is given by
\begin{equation}\label{Def Dyadic Indicator}
\boldsymbol{\mathbbm{1} } _{\mathcal{D} }(x) \defining \sum_{n \, \in \, \boldsymbol{\mathbbm{N} } }
\sum_{k \, = \, 1}^{ 2^{n - 1} }
\boldsymbol{\mathbbm{1} } _{\big\{\tfrac{2k - 1}{2^{n}}\big\}} (x) .
\end{equation}
\end{subequations}
\end{example}
\begin{corollary}\label{Th dyadic microstructure}
Let $\mathcal{D} $ be the Dyadic microstructure, then $V_{\mathcal{D} } = \{0\}$.
\end{corollary}
Next, we analyze a particular non-dense microstructure, which is a perfect set.
\begin{example}\label{Def Triadic Extremes Microstructure}
Let $\boldsymbol{C}$ be the Cantor set and $\boldsymbol{D}  \defining (0,1) - \boldsymbol{C}$ be its complement.
Let $\mathscr{C} $ be the set of extremes of the intervals removed in the construction of the Cantor set. We construct an $n$-stage development of $\mathscr{C} $ by storing recursively the new extremes of each stage in the construction of the Cantor set by removal of intervals
\begin{align}
\begin{split}
\mathpzc{F}_{1} \defining \Big\{\frac{1}{3},\, \frac{2}{3} \Big\}&  , \\
\mathpzc{F}_{n+1}\defining \Big\{\frac{1}{3}\, \mathpzc{c} : \mathpzc{c} \in \mathpzc{F}_{n}   \Big\} \cup
\Big\{1 - \frac{1}{3}\, \mathpzc{c} : \mathpzc{c} \in \mathpzc{F}_{n}   \Big\} &,\quad \forall \, n\geq 1 .
\end{split}
\end{align}
The microstructure of extremes $\mathscr{C} $ is given by
\begin{subequations}\label{Def Triadic Extremes Numbers}
\begin{equation}\label{Def Triadic Extremes Set}
\mathscr{C}  \defining \bigcup_{n \, \in \, \boldsymbol{\mathbbm{N} } }\mathpzc{F}_{n}  .
\end{equation}
Its indicator function is given by
\begin{equation}\label{Def Triadic Extremes Indicator}
\boldsymbol{\mathbbm{1} } _{\mathscr{C} }(x) \defining \sum_{n \, \in \,  \boldsymbol{\mathbbm{N} } }\boldsymbol{\mathbbm{1} } _{\mathpzc{F}_{n} } (x) .
\end{equation}
The $n$-stage partial development is given by
\begin{equation}\label{Def Triadic Extremes n stage development}
\mathcal{C}_{n}  \defining \bigcup_{i\, = \,1}^{n} \mathpzc{F}_{i} .
\end{equation}
\end{subequations}
\end{example}
Next, we find the strong problem whose weak formulation is given by \eqref{Pblm Variational Microstructure Concise} with the particular microstructure $\mathscr{C} $ defined above.
\begin{theorem}\label{Th strong limit problem}
Let $\{I_{n}:n\in \boldsymbol{\mathbbm{N} } \}$ be a enumeration of the connected components of $\boldsymbol{D} $ then, for each $n\in \boldsymbol{\mathbbm{N} } $, $p\vert_{I_{n}}$ is the unique solution of the problem
\begin{subequations}\label{Eq decoupling of limit problem}
\begin{equation}\label{Eq local limit laplacian}
-\boldsymbol{\partial } ^{2} \xi = F \quad \text{in} \; \;L^{2}(I_{n}) ,
\end{equation}
\begin{equation}\label{Eq ocal limit interface}
\xi = 0  \quad \text{on} \;\; \boldsymbol{\partial }  I_{n}.
\end{equation}
\end{subequations}
Moreover, the problem \eqref{Pblm Variational Microstructure Concise} is the weak formulation of the following strong problem with fractal microstructure $\mathscr{C} $
\begin{subequations}\label{Eq strong limit problem}
\begin{equation}\label{Eq limit laplacian}
-\boldsymbol{\partial } ^{2} p = F \quad \text{in} \; \;L^{2}(0,1) ,
\end{equation}
\begin{equation}\label{Eq limit interface}
p = 0  \quad \text{on} \;\; \mathscr{C}  ,
\end{equation}
\begin{equation}\label{Eq limit boundary conditions}
p(0)= 0  \, , \quad  p(1) = 0.
\end{equation}
\end{subequations}
\begin{proof}
Let $I_{n}$ be a fixed connected component of $\boldsymbol{D} $. Let $\varphi\in C_{0}^{\infty}(I_{n})$, notice that $\varphi \in V_{ \C}$ since $\varphi\vert_{\mathscr{C} } = 0$. Testing \eqref{Pblm Variational Microstructure Concise} with $\varphi\in C_{0}^{\infty}(I_{n})$ arbitrary we get
\begin{equation} \label{Eq testing on connected components}
\int_{I_{n}}\boldsymbol{\partial }  (p\vert_{I_{n}}) \, \boldsymbol{\partial }  \varphi =
\int_{I_{n}}\boldsymbol{\partial }  p \, \boldsymbol{\partial }  \varphi
= \int_{I_{n}}F\,\varphi\, , \quad \forall \; \varphi\in C_{0}^{\infty}(I_{n})
\end{equation}
Since $p\in V_{ \B}$ and $\mathscr{C} $ is a perfect set, then $p = 0$ on $\mathscr{C} $. In particular $p = 0$ on $\boldsymbol{\partial }  I_{n}$ i.e., $p\vert_{I_{n}}\in H_{0}^{1}(I_{n})$. Therefore, $p\vert_{I_{n}}$ is the unique solution to the weak variational formulation of \textit{Problem} \eqref{Eq decoupling of limit problem}. This finishes the first part. For the second part of the theorem integrate by parts the left hand side of the expression \eqref{Eq testing on connected components}, since the equality holds for any $\varphi\in C_{0}^{\infty}(I_{n})$, it follows that $-\boldsymbol{\partial } ^{2} p = F$ in $L^{2}(I_{n})$. Since $n\in \boldsymbol{\mathbbm{N} } $ was arbitrary we conclude $-\boldsymbol{\partial } ^{2} p = F$ in $L^{2}(\boldsymbol{D} )$. The equality \eqref{Eq limit laplacian} follows due to Cantor's set null Lebesgue measure. The interface statement given in equation \eqref{Eq limit interface} follows from the fact that $p\in V_{\mathscr{C} }$ and $\mathscr{C} $ is a perfect set.
\end{proof}
\end{theorem}
The theorem above shows that the limiting problem \eqref{Pblm Variational Microstructure Concise} can be solved as a countable collection of independent problems, i.e., it is fully-decoupled. Therefore, there is no fluid exchange between two connected components, and we must conclude that the outflow coming from any component is totally absorbed by a part of the Cantor set. Clearly, this is not a desirable model since it annihilates the possibility of  fluid exchange through the microstructure: the  role of the microstructure is reduced to become the boundary where the phenomenon takes place. On the other hand the dyadic microstructure $\mathcal{D} $ trivializes the problem \eqref{Pblm Variational Microstructure Concise} as it was seen in \textit{Corollary} \ref{Th dyadic microstructure}. In the next section we take a different approach to overcome these difficulties.
\end{comment}
\section{The Fractal Scaling Model}\label{Dec Fractal Scaling Modeling}
The unsatisfactory conclusions shown in the previous section are, essentially, due to a physical fact assumed in the model: that the storage fluid exchange coefficient $\beta$ is constant all over the microstructure $\mathsf{B} $. Consequently, the storage effect across the microstructure adds up to infinity. Hence, the modeling of $\beta$ has to avoid this hypothesis. On one hand, we need to assure that the form $\Lambda: V\times V\rightarrow \boldsymbol{\mathbbm{R} } $, defined by
\begin{equation}\label{Def Bilinear Form Scaled Coefficient}
\Lambda(q, r)\defining
\sum_{\mathpzc{b}  \, \in  \, \mathsf{B} } \beta(\mathpzc{b} ) \, q  (\mathpzc{b} ) \,  r(\mathpzc{b} ),
\end{equation}
is bilinear and continuous. On the other hand, \textit{Lemma} \ref{Th microstructure accumulation points} states the need to avoid global estimates for $\{p_{n} (\mathpzc{b} )\boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n} }(\mathpzc{b} ): \mathpzc{b} \in \mathsf{B}  \}\subseteq \ell^{t}(\mathsf{B} )$ for any $t > 1$. Therefore, if $\{\beta(\mathpzc{b} ): \mathpzc{b} \in \mathsf{B} \} \in \ell^{1}(\mathsf{B} )$ and recalling that $\beta(x) \geq 0$ for all $x\in (0,1)$, the bilinear form satisfies
\begin{equation} \label{Def Bilinear Form Scaled Coefficient Continuity}
\begin{split}
\bigg\vert  \sum_{\mathpzc{b}  \, \in  \, \mathsf{B} } \beta(\mathpzc{b} ) \,  q  (\mathpzc{b} )  \,  r(\mathpzc{b} ) \bigg\vert
&  \leq \sup_{\mathpzc{b} \,\in\,\mathsf{B} } \vert  q  (\mathpzc{b} ) \vert  \,  \sup_{\mathpzc{b} \,\in\,\mathsf{B} } \vert r(\mathpzc{b} ) \vert  \sum_{\mathpzc{b}  \, \in  \, \mathsf{B} }  \beta(\mathpzc{b} )  \\
 %
& \leq \Vert q \Vert_{V} \, \Vert r \Vert_{V} \sum_{\mathpzc{b}  \, \in  \, \mathsf{B} }  \beta(\mathpzc{b} ) .
 \end{split}
\end{equation}
In order to attain this condition, it is natural to assume that the storage coefficient $\beta$, scales consistently with the properties of the fractal microstructure. This motivates the following definition
\begin{definition}\label{Def scaled storage coefficient}
Let $\mathsf{B} \subseteq [0,1]$ be a fractal microstructure with $L > 1$ as given in \textit{Definition} \ref{Def Fractal Microstructure}. Then, a storage coefficient $\beta: \mathsf{B} \rightarrow (0, \infty)$ is said to \textbf{scale consistently} with a given $\sigma$-finite development $\{\mathsf{B}_{n} : n\geq 0 \}$, if it satisfies
\begin{equation}\label{Eq scaled storage coefficient}
\beta (\mathpzc{b} ) \defining  a \sum_{n \, \in \, \boldsymbol{\mathbbm{N} } } \Big(\frac{1}{L} - \epsilon \Big)^{n}
\, \boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n}  - \mathsf{B} _{n-1}}(\mathpzc{b} )\, ,
\quad \text{with}\; a > 0 \; \text{and}\;\,  0 < \epsilon<  \frac{1}{L} \, .
\end{equation}
\end{definition}
\begin{proposition}\label{Th scaled storage coefficient convergence}
Let $\beta: \mathsf{B} \rightarrow (0, \infty)$ be a storage coefficient consistently scaled with $\mathsf{B} $ then $\beta\in \ell^{1}(\mathsf{B} )$.
\end{proposition}
\begin{proof}
Since $\{\mathsf{B}_{n} :n \geq 0\}$ is the $\sigma$-finite development of $\mathsf{B} $ with $\mathsf{B} _{0}\neq\emptyset$, the cardinality \textit{Identity} \eqref{Eq Cardinality Relationships} implies
\begin{align*}
\sum_{\mathpzc{b} \,\in\,\mathsf{B}_{n} }\vert \beta(\mathpzc{b} ) \vert =
\sum_{\mathpzc{b} \,\in\,\mathsf{B}_{n} } \beta(\mathpzc{b} )
& =
\sum_{k \, = \, 1}^{n} \; \sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{k}  - \mathsf{B}_{k -1} } \beta(\mathpzc{b} ) \\
& =
\sum_{k \, = \, 1}^{n} a \, L^{k-1}(L-1) \, \card(\mathsf{B} _{0}) \Big(\frac{1}{L} - \epsilon \Big)^{k}
= a   \, \card(\mathsf{B} _{0})\big(1 - \frac{1}{L}\big) \sum_{k \, = \, 1}^{n} \big(1 - \epsilon \, L\big)^{k}  .
\end{align*}
Since $\epsilon\in (0, \frac{1}{L})$, as stated in \textit{Definition} \ref{Def scaled storage coefficient}, the expression above is convergent and the result follows.
\qed
\end{proof}
\begin{remark}
It is immediate to see some variations of \textit{Definition} \ref{Def scaled storage coefficient} based on the geometric series properties, for instance if $\{\alpha_{n}: n\in \boldsymbol{\mathbbm{N} } \}\subseteq (0,\infty)$ is a sequence such that $\limsup\limits_{ n } \sqrt[n]{\alpha_{n}} < \frac{1}{L}$ then, consider
\begin{equation*}
\beta (\mathpzc{b} ) \defining   \sum_{n \, \in \, \boldsymbol{\mathbbm{N} } }
\alpha_{n}\, \boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n}  - \mathsf{B} _{n-1}}(\mathpzc{b} ) .
\end{equation*}
The storage coefficient defined above will also satisfy that $\beta\in \ell^{1}(\mathsf{B} )$ and permit the desired \textit{Estimate} \eqref{Def Bilinear Form Scaled Coefficient Continuity}. A more sophisticated variation includes probabilistic uncertainty for the values $\{\alpha_{n}: n \in \boldsymbol{\mathbbm{N} } \}$, whether on its decay rate or the distributed values of tolerance centered at the self-similarity parameter $L$. A very basic probabilistic version of the latter will be numerically illustrated in \textit{Section} \ref{Sec Numerical Experiments}.
\end{remark}
\subsection{The Limit Problem}
In the following it will be assumed that $F\in L^{2}(0,1)$.
For simplicity of notation we understand that both, the storage coefficient and the interface forcing term are defined on the whole domain $[0,1]$, with $\beta\big\vert_{(0,1) - \mathsf{B} } = f\big\vert_{(0,1) - \mathsf{B} } = 0$.  It is also assumed that the storage coefficient $\beta\big\vert_{\mathsf{B} }$ scales consistently with the fractal microstructure $\mathsf{B} $ and that the forcing term is summable $f\in \ell^{1}(\mathsf{B} )$.
\begin{theorem}\label{Th well posedness good coefficient problems}
The following problems are well-posed.
\begin{align}\label{Pblm Variational n stage Good Coefficients}
& p_{n}  \in V : &
& \int_{0}^{1}\boldsymbol{\partial }  p_{n}  \, \boldsymbol{\partial }  q
+  \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n} } \beta(\mathpzc{b} )\, p_{n}   (\mathpzc{b} ) \,  q(\mathpzc{b} )= \int_{0}^{1}F\,q
+ \sum_{\mathpzc{b}  \, \in \, \mathsf{B}_{n} } f(\mathpzc{b} ) \,  q(\mathpzc{b} )\,, &
& \forall q\in V.
\end{align}
\begin{align}\label{Pblm Variational Microstructure Good Coefficients}
& p \in V : &
& \int_{0}^{1}\boldsymbol{\partial }  p \, \boldsymbol{\partial }  q
+  \sum_{\mathpzc{b}  \, \in  \, \mathsf{B} } \beta(\mathpzc{b} )\, p (\mathpzc{b} ) \,  q(\mathpzc{b} )= \int_{0}^{1}F\,q
+ \sum_{\mathpzc{b}  \, \in \, \mathsf{B} } f(\mathpzc{b} ) \,  q(\mathpzc{b} )\,, &
& \forall q\in V.
\end{align}
\end{theorem}
\begin{proof}
Consider the bilinear forms
\begin{subequations} \label{Def Bilinear Forms Scaled Coefficient}
\begin{equation}\label{Def Bilinear Forms Scaled Coefficient n stage}
\langle q, r\rangle_{\scriptscriptstyle \mathsf{B}_{n} }  \defining
\int_{0}^{1}\boldsymbol{\partial }  q \, \boldsymbol{\partial }  r
+  \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n} } \beta(\mathpzc{b} )\, q  (\mathpzc{b} ) \,  r(\mathpzc{b} ) \, ,  \quad n\in \boldsymbol{\mathbbm{N} } ,
\end{equation}
\begin{equation}\label{Def Bilinear Forms Scaled Coefficient full Microstructure}
\langle q, r\rangle_{\scriptscriptstyle \mathsf{B} }  \defining \int_{0}^{1}\boldsymbol{\partial }  q \, \boldsymbol{\partial }  r
+  \sum_{\mathpzc{b}  \, \in  \, \mathsf{B} } \beta(\mathpzc{b} )\, q  (\mathpzc{b} ) \,  r(\mathpzc{b} ) .
\end{equation}
\end{subequations}
Since $\beta\in \ell^{1}(\mathsf{B} )$, the \textit{Inequality} \eqref{Def Bilinear Form Scaled Coefficient Continuity} is satisfied and the bilinear forms $(q, r)\mapsto \sum\limits_{\mathpzc{b}  \, \in  \, \mathsf{B} } \beta(\mathpzc{b} )\, q  (\mathpzc{b} ) \,  r(\mathpzc{b} )$, $(q, r)\mapsto \sum\limits_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n} } \beta(\mathpzc{b} )\, q  (\mathpzc{b} ) \,  r(\mathpzc{b} )$ for each $n\in \boldsymbol{\mathbbm{N} } $
are continuous; consequently the bilinear forms \eqref{Def Bilinear Forms Scaled Coefficient} are also continuous.  On the other hand, since the coefficient $\beta$ is non-negative the bilinear forms \eqref{Def Bilinear Forms Scaled Coefficient} are both coercive on $V$. Applying the Lax-Milgram Theorem the result follows.
\qed
\end{proof}
\begin{theorem}
Let $\{p_{n} : n\in \boldsymbol{\mathbbm{N} }  \}$ be the sequence of solutions of Problems \eqref{Pblm Variational n stage Good Coefficients} and let $p$ be the solution to Problem \eqref{Pblm Variational Microstructure Good Coefficients}. Then
\begin{enumerate}[(i)]
\item $\{p_{n} : n\in \boldsymbol{\mathbbm{N} }  \}$  converges weakly to $p$ in $V$.

\item The following convergence statements hold
\begin{subequations}\label{Stmt Convergence Interface Functionals}
\begin{equation} \label{Stmt Strong Convergence Non-Linear Interface Functionals}
\sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n} } \beta(\mathpzc{b} )\, p_{n} ^{2} (\mathpzc{b} ) \xrightarrow[n\rightarrow \infty]{}
\sum_{\mathpzc{b}  \, \in  \, \mathsf{B} } \beta(\mathpzc{b} )\, p^{2} (\mathpzc{b} ).
\end{equation}
\begin{equation} \label{Stmt Strong Convergence Forcing Interface Functionals}
\sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n} } f(\mathpzc{b} )\, p_{n}  (\mathpzc{b} ) \xrightarrow[n\rightarrow \infty]{}
\sum_{\mathpzc{b}  \, \in  \, \mathsf{B} } f(\mathpzc{b} )\, p (\mathpzc{b} ).
\end{equation}
\end{subequations}

\item $\{p_{n} : n\in \boldsymbol{\mathbbm{N} }  \}$  converges strongly to $p$ in $V$.
\end{enumerate}
\end{theorem}
\begin{proof}
\begin{enumerate}[(i)]
\item The result follows using the techniques presented in \textit{Theorem} \ref{Th boundedness of the sequence of solutions} yield the boundedness of the sequence $\{p_{n} : n\in \boldsymbol{\mathbbm{N} }  \}$ and using the reasoning of \textit{Theorem} \ref{Th weak convergence of the solutions in V} the weak convergence follows.

\item Due to the weak convergence of the solutions and the fact that the evaluation is a continuous linear functional, it follows that $p_{n} (\mathpzc{b} ) \rightarrow p(\mathpzc{b} )$ and $p_{n} ^{2}(\mathpzc{b} ) \rightarrow p^{2}(\mathpzc{b} )$ for all $\mathpzc{b} \in \mathsf{B} $. Let $M> 0$ be such that $\Vert p \Vert_{V}^{2}\leq M$ and $\Vert p_{n}  \Vert_{V}^{2}\leq M$ for all $n\in \boldsymbol{\mathbbm{N} } $. Fix $k\in \boldsymbol{\mathbbm{N} } $ such that $n > k$ implies $\Big\vert \sum\limits_{\mathsf{B}  - \mathsf{B}_{n}  } \beta(\mathpzc{b} )\Big\vert \leq \dfrac{\epsilon}{3\, M^{2}}$, which we know to exist since $\beta\in \ell^{1}(\mathsf{B} )$. Then, for any $n > k$ it holds that
\begin{equation*}
\begin{split}
\bigg\vert \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n} } \beta(\mathpzc{b} )\, p_{n} ^{2} (\mathpzc{b} ) -
\sum_{\mathpzc{b}  \, \in  \, \mathsf{B} }  & \beta(\mathpzc{b} )\, p^{2} (\mathpzc{b} ) \bigg\vert \\
& \leq
\sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{k} } \beta(\mathpzc{b} )\,\left\vert p_{n} ^{2} (\mathpzc{b} ) -
 p^{2} (\mathpzc{b} ) \right\vert +
 \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{n}  - \mathsf{B}_{k} } \beta(\mathpzc{b} )\, \left\vertp_{n} ^{2} (\mathpzc{b} ) \right\vert
 %
+  \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}  - \mathsf{B}_{k} } \beta(\mathpzc{b} )\, \left\vert
p^{2} (\mathpzc{b} ) \right\vert\\
& \leq
\sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{k} } \beta(\mathpzc{b} )\,\left\vert p_{n} ^{2} (\mathpzc{b} ) -p^{2} (\mathpzc{b} ) \right\vert
+ \Vert
 p_{n} \Vert_{V}^{2}    \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}  - \mathsf{B}_{k} } \beta(\mathpzc{b} )\,
 + \Vert
   p\Vert_{V}^{2} \sum_{\mathpzc{b}  \, \in  \, \mathsf{B}  - \mathsf{B}_{k} } \beta(\mathpzc{b} ) \\
& \leq
\sum_{\mathpzc{b}  \, \in  \, \mathsf{B}_{k} } \beta(\mathpzc{b} )\,\left\vert p_{n} ^{2} (\mathpzc{b} ) -p^{2} (\mathpzc{b} ) \right\vert
+ \frac{2}{3}\, \epsilon . \\
\end{split}
\end{equation*}
Since $k\in \boldsymbol{\mathbbm{N} } $ is fixed choose $N\in \boldsymbol{\mathbbm{N} } $ such that $n\geq N$ implies
\begin{equation*}
\vert p_{n} ^{2}(\mathpzc{b} ) - p^{2}(\mathpzc{b} ) \vert \leq \frac{\epsilon}{3} \;
\big\Vert \beta \big\Vert_{\ell^{1}(\mathsf{B} )}^{-1}
\quad \forall \; \mathpzc{b} \in \mathsf{B}_{k} .
\end{equation*}
Hence, combining with the previous estimate, the convergence \textit{Statement} \eqref{Stmt Strong Convergence Non-Linear Interface Functionals} follows. For the \textit{Statement} \eqref{Stmt Strong Convergence Forcing Interface Functionals} it is enough to combine the strong convergence of the forcing terms $\big\Vert f\boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n} } -f\big\Vert_{\ell^{1}(\mathsf{B} )}\xrightarrow[n \, \rightarrow \, \infty]{}0$ with the weak convergence of the solutions $p_{n}  \overset{w}\rightharpoonup p$. This completes the second part.

\item Again, the result follows combining the convergence \textit{Statements} \eqref{Stmt Strong Convergence Non-Linear Interface Functionals} and \eqref{Stmt Strong Convergence Forcing Interface Functionals} , with the techniques presented in \textit{Lemma} \ref{Th norms convergence} used to attain the strong convergence of the solutions.
\qed
\end{enumerate}
\end{proof}
\begin{comment}
Before finding the ``strong form" of \textit{Problem} \eqref{Pblm Variational Microstructure Good Coefficients} we need to prove a previous result and introduce some definitions and notation. For any function $q\in V$ we denote its support by $\supp (q)  = \cl \{x\in (0,1): q(x) = 0 \}$.
\begin{definition}\label{Def smooth mollifiers}
For any $x\in \boldsymbol{\mathbbm{R} } $, $\epsilon > 0$ and $\rho \in C_{0}^{\infty}(\boldsymbol{\mathbbm{R} } )$, denote by $\rho_{\epsilon}, T_{x} \rho: \boldsymbol{\mathbbm{R} } \rightarrow \boldsymbol{\mathbbm{R} } $ the translation an dilation associated functions i.e.,
\begin{align}\label{Eq translation and dilation}
& (T_{ x}  \rho)(y) \defining \rho (y - x) \, , &
&  \rho_{ \epsilon } (y) \defining \rho \big(\frac{y}{\epsilon}\big) \, , &
& \forall\, y\in \boldsymbol{\mathbbm{R} }  .
\end{align}
For any $x\in (0,1)$ define the set
\begin{equation}\label{Eq local smooth mollifiers}
\mathcal{H}_{ x } \defining \big\{ \rho \in C_{0}^{\infty}(\boldsymbol{\mathbbm{R} } ):T_{ x } \, \rho \in V \big\} =
\big\{ \rho \in C_{0}^{\infty}(\boldsymbol{\mathbbm{R} } ): \supp(T_{ x } \, \rho) \subseteq (0,1) \big\} .
\end{equation}
Finally, define the set
\begin{equation}\label{Eq smooth mollifiers}
\mathcal{H} \defining \left\{ \rho \in C_{0}^{\infty}(\boldsymbol{\mathbbm{R} } ): \rho (0) = 1 \right\} .
\end{equation}
\end{definition}
\begin{remark}
Clearly if $T_{ x }  \, \rho\in \mathcal{H}_{x}$ then $(T_{\!x } \, \rho)_{\epsilon} \in \mathcal{H}_{x}$ for all $\epsilon > 0$. Moreover, for any $x\in (0,1)$ and $\rho\in C_{0}^{\infty}(\boldsymbol{\mathbbm{R} } )$, in particular if $\rho\in \mathcal{H}$, there exists $\epsilon_{0} = \epsilon_{0}(x, \rho)> 0$ such that $0 < \epsilon < \epsilon_{0} $ implies that $(T_{x } \, \rho)_{\epsilon} \in \mathcal{H}_{x}$.
\end{remark}
In the following $\mathsf{B} '$ indicates the set of limit points of $\mathsf{B} $. The next result is devoted only to analyze the behavior of \textit{Problem} \eqref{Pblm Variational Microstructure Good Coefficients} on the limit points of the microstructure.
\begin{lemma}\label{Th non localizable operator for limit points}
\begin{enumerate}[(i)]
\item Let $\mathsf{b}_{0}  \in \mathsf{B} \cap \mathsf{B} '$ then
\begin{equation} \label{Eq localization limit interface accumulation points}
\lim_{\epsilon\, > \,0} \frac{1}{\epsilon} \int_{0}^{1} \boldsymbol{\partial }  p \, (T_{ \bnot} \boldsymbol{\partial } \rho)_{\epsilon}  \,
+  \beta(\mathsf{b}_{0} )\, p (\mathsf{b}_{0} ) =
f(\mathsf{b}_{0} ) \, ,  \quad \forall\, \rho \in \mathcal{H} .
\end{equation}

\item Let $\mathsf{b}_{0}  \in \mathsf{B} '- \mathsf{B} $ then
\begin{equation}
\lim_{\epsilon\, > \,0} \frac{1}{\epsilon} \int_{0}^{1} \boldsymbol{\partial }  p \, (T_{ \bnot} \boldsymbol{\partial } \rho)_{\epsilon}  \,
= 0 \, ,  \quad \forall\, \rho \in \mathcal{H} .
\end{equation}
\end{enumerate}
In the statements above, it is understood that the limit is computed for values of $\epsilon> 0$, small enough to assure that $(\Tnot\, \rho)_{ \epsilon} \in \mathcal{H}_{\mathsf{b}_{0} }$.
\end{lemma}
\begin{proof}
\begin{enumerate}[(i)]
\item
Let $\mathsf{b}_{0} \in \mathsf{B} \cap \mathsf{B} '$, $\rho \in \mathcal{H}$ and $\epsilon_{0} = \epsilon_{0}(\mathsf{b}_{0} , \rho)$ such that $0< \epsilon < \epsilon_{0}$ implies that $(\Tnot\, \rho)_{ \epsilon}\in \mathcal{H}_{\mathsf{b}_{0} }$; we denote $U_{ \epsilon} = \supp (\Tnot\, \rho)_{ \epsilon} $. Now test \textit{Problem} \eqref{Pblm Variational Microstructure Good Coefficients} with $(\Tnot\, \rho)_{ \epsilon}$ to get
\begin{equation} \label{Eq Eq first test with rho dilated}
\int_{U_{ \epsilon}} \boldsymbol{\partial }  p \, \boldsymbol{\partial }  \big[(\Tnot\, \rho)_{ \epsilon} \big]
+  \sum_{\mathpzc{b}  \, \in  \,  U_{ \epsilon} \cap \mathsf{B}  } \beta(\mathpzc{b} )\, p (\mathpzc{b} ) \,  (\Tnot\, \rho)_{ \epsilon}(\mathpzc{b} ) =
\int_{U_{ \epsilon}} F\,(\Tnot\, \rho)_{ \epsilon}
+ \sum_{\mathpzc{b}  \, \in  \, U_{ \epsilon} \cap \mathsf{B}  }  f(\mathpzc{b} ) \,  (\Tnot\, \rho)_{ \epsilon}(\mathpzc{b} ) .
\end{equation}
The equality $\boldsymbol{\partial }  \left[(\Tnot\, \rho)_{ \epsilon} \right] = (T_{ \bnot} \boldsymbol{\partial } \rho)_{\epsilon}$ is trivial. Notice that $\Vert (\Tnot\, \rho)_{ \epsilon} \Vert_{L^{2}(0,1)}\rightarrow 0$, therefore the first summand on the left hand side converges to zero as $\epsilon \rightarrow 0$. For the series on the right hand side we proceed as follows, since $f \in \ell^{1}(\mathsf{B} )$, for any $\delta > 0$ there exists $N\in \boldsymbol{\mathbbm{N} } $ such that $n> N$ implies
\begin{equation}\label{Ineq using L 1 properties on the interface forces}
\sum_{\mathpzc{b} \, \in \mathsf{B}  - \mathsf{B}_{n} } \vert f(\mathpzc{b} )\vert <
\frac{\delta}{\Vert \, \rho \Vert_{L^{\infty}(0,1)}} \,  .
\end{equation}
Define $d  = \min \{ \vert \mathsf{b}_{0}  - \mathpzc{b} \vert : \mathpzc{b}  \in \mathsf{B}_{n}  - \{ \mathsf{b}_{0} \} \}$, then $U_{ \epsilon} \cap \mathsf{B}_{n} \ = \{\mathsf{b}_{0} \}$ for all $\epsilon \in (0 , \min\{d, \epsilon_{0}\})$. Then
\begin{multline*}
\begin{split}
\bigg\vert f(\mathsf{b}_{0} ) - \sum_{\mathpzc{b} \, \in U_{ \epsilon} \cap \mathsf{B}  }  f(\mathpzc{b} ) (\Tnot\, \rho)_{ \epsilon} (\mathpzc{b} )\bigg\vert & =
\bigg\vert f(\mathsf{b}_{0} ) - \sum_{\mathpzc{b} \, \in \, U_{ \epsilon}\cap \mathsf{B}_{n} }  f(\mathpzc{b} ) (\Tnot\, \rho)_{ \epsilon} (\mathpzc{b} ) -
\sum_{\mathpzc{b} \, \in U_{ \epsilon}\cap (\mathsf{B}  - \mathsf{B}_{n} ) }  f(\mathpzc{b} ) (\Tnot\, \rho)_{ \epsilon} (\mathpzc{b} ) \bigg\vert \\
& = \bigg\vert
\sum_{\mathpzc{b} \, \in U_{ \epsilon}\cap (\mathsf{B}  - \mathsf{B}_{n} ) }  f(\mathpzc{b} ) (\Tnot\, \rho)_{ \epsilon} (\mathpzc{b} ) \bigg\vert \\
&
\leq
\sum_{\mathpzc{b} \, \in U_{ \epsilon}\cap (\mathsf{B}  - \mathsf{B}_{n} ) }  \vert f(\mathpzc{b} ) \vert \; \vert(\Tnot\, \rho)_{ \epsilon} (\mathpzc{b} )\vert \\
& \leq \sum_{\mathpzc{b} \, \in \mathsf{B}  - \mathsf{B}_{n}  }  \vert f(\mathpzc{b} ) \vert \; \Vert(\Tnot\, \rho)_{ \epsilon} \Vert_{L^{\infty}(0,1)}
= \Vert\, \rho \Vert_{L^{\infty}(0,1)}  \sum_{\mathpzc{b} \, \in \mathsf{B}  - \mathsf{B}_{n}  }  \vert f(\mathpzc{b} ) \vert < \delta .
\end{split}
\end{multline*}
The equality of the second line holds because $U_{ \epsilon} \cap \mathsf{B}_{n} \ = \{\mathsf{b}_{0} \}$ and $(\Tnot\, \rho)_{ \epsilon} (\mathsf{b}_{0} ) = 1$. The equality of the fourth line holds since $\Vert\, \rho \Vert_{L^{\infty}(0,1)} = \Vert\, (\Tnot\, \rho)_{ \epsilon} \Vert_{L^{\infty}(0,1)}$. The last inequality holds due to \textit{Estimate} \eqref{Ineq using L 1 properties on the interface forces}. Since the above holds for $\delta> 0$ arbitrary we conclude that $\sum_{\mathpzc{b} \, \in U_{ \epsilon} }  f(\mathpzc{b} ) (\Tnot\, \rho)_{ \epsilon} (\mathpzc{b} ) $ converges to $f(\mathsf{b}_{0} )$ as $\epsilon \rightarrow 0$. The convergence of the series $\sum_{\mathpzc{b}  \, \in  \, U_{ \epsilon}} \beta(\mathpzc{b} )\, p (\mathpzc{b} ) \,  (\Tnot\, \rho)_{ \epsilon}(\mathpzc{b} )$ to $\beta(\mathsf{b}_{0} )\, p(\mathsf{b}_{0} )$ is shown using the previous reasoning because $\beta\in \ell^{1}(\mathsf{B} )$ and therefore all the hypotheses hold. Thus, three out of four terms in the \textit{Equality} \eqref{Eq Eq first test with rho dilated} converge. It follows that the last summand is also convergent, i.e.
\begin{equation*}
\lim_{\epsilon\, > \,0} \frac{1}{\epsilon} \int_{U_{ \epsilon}} \boldsymbol{\partial }  p \, (T_{ \bnot} \boldsymbol{\partial } \rho)_{\epsilon}
= \lim_{\epsilon\, > \,0}  \int_{U_{ \epsilon}} \boldsymbol{\partial }  p \, \boldsymbol{\partial }  \left[ (\Tnot\, \rho)_{ \epsilon} \right]
=
f(\mathsf{b}_{0} ) -   \beta(\mathsf{b}_{0} )\, p (\mathsf{b}_{0} ) .
\end{equation*}

\item This follows immediately using the previous arguments with the microstructure $\mathsf{B}_{0}  \defining \mathsf{B}  \cup \{\mathsf{b}_{0}  \}$ and defining extensions $f_{0}, \beta_{0}$ of $f, \beta$ to $\mathsf{B}_{0} $ in the trivial way i.e. $f_{0}(\mathsf{b}_{0} ) = \beta_{0}(\mathsf{b}_{0} ) = 0$.
\qed
\end{enumerate}
\end{proof}
\begin{remark}
\begin{enumerate}[(i)]
\item
It is important to stress that the technique of integration by parts can not be applied to \textit{Equality}  \eqref{Eq Eq first test with rho dilated} if $\mathsf{b}_{0} \in \mathsf{B}  '$ in order to conclude that $-\boldsymbol{\partial } ^{2} p \in L^{2}(U_{ \epsilon})$. The reason is because for each $\epsilon>0$, the neighborhood $U_{ \epsilon}$ contains infinitely elements of $\mathsf{B} $ and both series in the expression hit infinitely many summands, which are jumps.

\item
Notice that, defining $U = \supp(\rho)$, the following identities hold
\begin{equation*}
\lim_{\epsilon\, > \,0} \frac{1}{\epsilon} \int_{U_{ \epsilon}} \boldsymbol{\partial }  p \, (T_{ \bnot} \boldsymbol{\partial } \rho)_{\epsilon}
= \lim_{\epsilon\, > \,0} \frac{\vert U_{ \epsilon} \vert}{\epsilon} \, \frac{1}{\vert U_{ \epsilon} \vert}\int_{U_{ \epsilon}} \boldsymbol{\partial }  p \, (T_{ \bnot} \boldsymbol{\partial } \rho)_{\epsilon}  = \lim_{\epsilon\, > \,0} \frac{\epsilon \, \vert U \vert}{\epsilon} \, \frac{1}{\vert U_{ \epsilon} \vert}\int_{U_{ \epsilon}} \boldsymbol{\partial }  p \, (T_{ \bnot} \boldsymbol{\partial } \rho)_{\epsilon}.
\end{equation*}
However, Lebesgue's differentiation theorem can not be used to conclude that the last expression is equal to $\vert U \vert \, \boldsymbol{\partial }  p (\mathsf{b}_{0} )\, \boldsymbol{\partial }  \rho(0)$ because the last integrand also depends on the limit variable $\epsilon$.

\item
The observations above, show that for points $\mathsf{b}_{0}  \in \mathsf{B}  '$ no point-wise statements can be achieved from the limiting \textit{Problem} \eqref{Pblm Variational Microstructure Good Coefficients}. The limiting operator is non-localizable on these points.
\end{enumerate}
\end{remark}
With the discussion above we are ready to prove the following result.
\end{comment}
Next, we present the closest version of a strong form of \textit{Problem} \eqref{Pblm Variational Microstructure Good Coefficients}.
\begin{theorem}\label{Th pseudo strong form}
Let $\mathsf{B} \subseteq [0,1]$ be a microstructure and let $\mathsf{B} '$ be its set of limit points.
Then, \textit{Problem} \eqref{Pblm Variational Microstructure Good Coefficients} is a weak formulation of the following strong problem.
\begin{subequations}\label{Pblm Strong Microstructure Good Coefficients}
\begin{equation} \label{Eq limit laplacian Microstructure Good Coefficients}
-\boldsymbol{\partial } ^{2} p
= F \quad  \text{in the sense } \; \;L^{2} \big((0,1) - \mathsf{B}  '
\big) ,
\end{equation}
\begin{equation} \label{Eq boundary conditions limit problem}
p (0) = 0 .
\end{equation}
Together with the interface fluid transmission conditions for isolated points of $\mathsf{B} $
    \begin{equation}\label{Eq strong limit interface isolated points}
    %
    \begin{split}
    p \left(\mathsf{b}_{0} ^{-}\right) & = p\left(\mathsf{b}_{0} ^{+}\right)  ,
    \\

    \boldsymbol{\partial }  p \left( \mathsf{b}_{0} ^{-}\right)  -  \boldsymbol{\partial }  p \left( \mathsf{b}_{0} ^{+}\right) +
   \beta (\mathsf{b}_{0} ) \,  p \left( \mathsf{b}_{0} \right) & =  f\left( \mathsf{b}_{0} \right)
    \quad \forall\, \mathsf{b}_{0}  \in \mathsf{B}  - \mathsf{B} '.
    \end{split}
    \end{equation}
And the non-localizable fluid transmission conditions for limit points of $\mathsf{B} $
\begin{equation}\label{Eq strong limit interface accumulation points extra hypothesis}
\begin{split}
 p \left(x^{-}\right) & = p\left(x^{+}\right)  ,
    \\
\lim\limits_{n \, \rightarrow \, \infty}
\frac{1}{\frac{1}{n}}
\int_{x - \frac{1}{n}}^{x + \frac{1}{n}}\boldsymbol{\partial }  p
+  \beta(x)\, p (x) \boldsymbol{\mathbbm{1} } _{\mathsf{B} }(x) & =
f(x) \, .
\end{split}
\end{equation}

\end{subequations}
\end{theorem}
\begin{proof}
The boundary condition  \eqref{Eq boundary conditions limit problem} holds because $p \in V$. Let $x\in (0,1) - \mathsf{B}  '$, then, there exists $\delta_{0}> 0 $ such that $(x - \delta, x+ \delta)\cap \mathsf{B}  = \{x\}\cap \mathsf{B} $ for all $\delta \in (0, \delta_{0})$. Test \textit{Problem} \eqref{Pblm Variational Microstructure Good Coefficients} with $\varphi\in C_{0}^{\infty}(x-\delta, x)$ to get
\begin{equation*}
-\big\langle \boldsymbol{\partial } ^{2} p, \, \varphi\big\rangle_{\mathcal{D}  '(x - \delta, x), D(x - \delta, x)}
= \int_{x - \delta}^{x }   \boldsymbol{\partial }  p \, \boldsymbol{\partial }  \varphi  =
\int_{x - \delta}^{x } F\varphi .
\end{equation*}
Since the above holds for each smooth function on $(x - \delta, x)$, we conclude that $- \boldsymbol{\partial } ^{2} p = F$ in $L^{2}(x - \delta, x)$. Similarly, it follows that $- \boldsymbol{\partial } ^{2} p = F$ in $L^{2}(x, x + \delta)$, therefore $\boldsymbol{\partial }  p\in H^{1}(x - \delta, x)\cap H^{1}(x, x+ \delta)$ and the statement \eqref{Eq limit laplacian Microstructure Good Coefficients} follows. Now, choose $\varphi \in C_{0}^{\infty}(x - \delta, x + \delta)$ and test \textit{Problem} \eqref{Pblm Variational Microstructure Good Coefficients} to get
\begin{equation*}
\int_{x - \delta}^{x + \delta}   \boldsymbol{\partial }  p \, \boldsymbol{\partial }  \varphi
+\beta(x)\, p(x) \, \varphi(x)
= \int_{x - \delta}^{x + \delta} F\varphi
+ f(x) \, p(x)  \, \varphi(x) .
\end{equation*}
Next, in the expression above, we break the interval conveniently in order to use integration by parts
\begin{equation*}
\begin{split}
\int_{x - \delta}^{x + \delta} F\varphi
+ f(x) \, p(x)  \, \varphi(x)
& = \int_{x - \delta}^{x }   \boldsymbol{\partial }  p \, \boldsymbol{\partial }  \varphi
+ \int_{x }^{x + \delta}   \boldsymbol{\partial }  p \, \boldsymbol{\partial }  \varphi
+\beta(x)\, p(x)  \, \varphi(x)  \\
& = - \int_{x - \delta}^{x }\boldsymbol{\partial } ^{2} p \, \varphi
- \int_{x }^{x + \delta}   \boldsymbol{\partial } ^{2} p \,  \varphi
+ \boldsymbol{\partial }  p\, \varphi \big\vert_{x - \delta}^{x}
+ \boldsymbol{\partial }  p\, \varphi \big\vert_{x }^{x + \delta}
+\beta(x)\, p(x)  \, \varphi(x)  .
\end{split}
\end{equation*}
From the initial part of the proof, the first summand of the left hand side cancels with the first two summands of the right hand side. Evaluating the boundary terms and recalling that $\varphi (x - \delta) = \varphi(x + \delta) = 0$ this gives
\begin{equation*}
f(x) = \boldsymbol{\partial }  p(x^{-})
- \boldsymbol{\partial }  p(x^{+})
+   \beta(x)\, p (x).
\end{equation*}
Consequently, the normal flux balance condition in \eqref{Eq strong limit interface isolated points} follows for points $x\in \mathsf{B}  -\mathsf{B} '$. The normal stress balance condition in \eqref{Eq strong limit interface isolated points} follows from the continuity of $p$ across the interface which holds for any function in $V\subseteq H^{1}(0,1)$. The normal stress balance in \textit{Statement} \eqref{Eq strong limit interface accumulation points extra hypothesis} holds due to the continuity of $p$ at any point of $x\in (0,1)$. Now, fix $x\in \mathsf{B} '$ and test the \textit{Problem} \eqref{Pblm Variational Microstructure Good Coefficients} with $q_{n}(t)\defining q\big(n(t - x)\big)$, where $q(t) \defining (t +1)\boldsymbol{\mathbbm{1} } _{(-1,0)}(t) + (1 - t)\boldsymbol{\mathbbm{1} } (0,1)(t)$; this gives
\begin{equation*}
\frac{1}{\frac{1}{n}}\int_{x - \frac{1}{n}}^{x+\frac{1}{n}}\boldsymbol{\partial }  p
+  \sum_{\mathpzc{b}  \, \in  \, \mathsf{B} } \beta(\mathpzc{b} )\, p (\mathpzc{b} ) \,  q_{n}(\mathpzc{b} )
= \int_{x - \frac{1}{n}}^{x+\frac{1}{n}} F\,q_{n}
+ \sum_{\mathpzc{b}  \, \in \, \mathsf{B} } f(\mathpzc{b} ) \,  q_{n}(\mathpzc{b} ) .
\end{equation*}
Taking limits in the expression above and recalling the Lebesgue Dominated Convergence Theorem the result follows.
\qed
\end{proof}
Notice that, by definition of distributions, the following equality holds in the sense of distribution for the solution of \textit{Problem} \eqref{Pblm Variational Microstructure Good Coefficients}
\begin{equation*}
-\boldsymbol{\partial } ^{2} p + \sum_{\mathpzc{b}  \, \in  \, \mathsf{B} } \beta(\mathpzc{b} )\, p (\mathpzc{b} ) \,  \delta_{\{\mathpzc{b} \}}
= F + \sum_{\mathpzc{b}  \, \in \, \mathsf{B} } f(\mathpzc{b} ) \delta_{\{\mathpzc{b} \}}
 \, , \quad
\text{in}
\; \;\mathcal{D}  '  (0, 1) .
\end{equation*}
Where $\delta_{\{\mathpzc{b} \}}(q) \defining q(\mathpzc{b} )$ is the Dirac evaluation functional.
It is easy to observe that if $x$ is a limit point of $\mathsf{B} $, both series in the expression above will pick up infinitely many non-null terms for test functions such that $\supp(\varphi)\cap \mathsf{B} ' \neq \emptyset$. Therefore, the techniques used in \textit{Theorem} \ref{Th pseudo strong form} to derive point-wise statements from weak variational ones, do not apply. Again, this fact is unfortunate because, as pointed out at the end of \textit{Section} \ref{Sec A sigma finite Interface Microstructure}, several important fractal microstructures are dense in $[0,1]$ or perfect sets. If $\mathsf{B} $ is dense very little can be said about the point-wise behavior at any point of the domain and if $\mathsf{B} $ is a perfect set, the transmission conditions can not be described locally at any point of the interface. A positive aspect of this model is that the aforementioned cases do not become trivial or uninteresting. Additionally, if the set of accumulation points $\mathsf{B}  '$ has null Lebesgue measure the non-localizable interface conditions are not relevant for the global phenomenon. Fortunately, this is the case for very important fractal microstructures e.g., the collection of extremes of the removed intervals in the construction of the Cantor set, the Sierpinski Triangle in 2-D, etc.
\section{Numerical Experiments}\label{Sec Numerical Experiments}
In this section we present two types of numerical experiments. The first type are verification examples, supporting our homogenization conclusions for a problem whose asymptotic behavior is known exactly.  The second type are of exploratory nature, in order to gain heuristic understanding of the probabilistic variations of the model. The experiments are executed in a MATLAB code using the Finite Element Method (FEM); it is an adaptation of the code \textbf{fem1d.m} \cite{PeszynskaFEM}.
\subsection{General Setting}
In all the examples to be presented below the following holds. For the sake of simplicity, the microstructure $\mathsf{B} \subseteq [0,1]$ is the collection of extremes of the removed intervals in the construction of the Cantor set. In addition, its $\sigma$-finite development $\{\mathsf{B}_{n} : n\geq 0 \}$ is the natural one i.e.,
\begin{align}\label{Def Partial Triadic Extremes Set}
\begin{split}
\mathsf{B} _{0} \defining \big\{0, 1 \big\}&  , \\
\mathsf{B} _{n+1}\defining \Big\{\frac{1}{3}\, \mathpzc{b} : \mathpzc{b} \in \mathsf{B}_{n}   \Big\} \cup
\Big\{1 - \frac{1}{3}\, \mathpzc{b} : \mathpzc{b} \in \mathsf{B}_{n}   \Big\} &,\quad \forall \, n\geq 1 .
\end{split}
\end{align}
For the experiments it will be shown that the sequences $\{p_{n} : n\in \boldsymbol{\mathbbm{N} }  \}$ are Cauchy; where $n\in \boldsymbol{\mathbbm{N} } $ indicates de stage of the Cantor set construction. In all the cases, the computations are made for the stages $6, 7,8, 9$. For each example we present graphics for values of an $n$-stage chosen from $\{3,4,6,9\}$, based on optical neatness; two or three graphs for the solution and the graphs of the corresponding derivative. Also, for visual purposes we introduce vertical lines in the graph of the derivatives, to highlight the fractal structure of the function. For all the examples we set the forcing term $F\in L^{2}(0,1)$ equal to zero and he interface forcing term $f: \mathsf{B} \rightarrow \boldsymbol{\mathbbm{R} } $, is given by the following expression
\begin{equation}\label{Def Deterministic Interface Forcing Term}
f (\mathpzc{b} ) \defining \sum_{n \, \in \, \boldsymbol{\mathbbm{N} } } \frac{1}{3^{n}}
\, \boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n}  - \mathsf{B} _{n-1}}(\mathpzc{b} )\, .
\end{equation}
\subsection{The Examples}
\begin{example}[Unscaled Example]\label{Ex Basic Example}
In the first example we implement the model studied in \textit{Section} \ref{Sec A sigma finite Interface Microstructure}, in order to verify its conclusions. In particular, it must hold that $\Vert p_{n} \Vert_{V} \rightarrow 0$. We set the storage coefficient $\beta = 1$ and the forcing terms $F = 0$, $f$ defined in \eqref{Def Deterministic Interface Forcing Term} above. The convergence table is displayed below, together with the corresponding graphics.
\begin{table*}[h!]
\centerline{\textit{Example} \ref{Ex Basic Example} : Convergence Table, $\beta = 1$.}\vspace{5pt}
\def1.4{1.4}
\begin{center}
\begin{tabular}{ c c c c c c }
    \hline
Stage $n$ &  $\#$ Nodes & $\Vert  p_{n}   \Vert_{L^{2}(0,1)} $ & $\Vert  p_{n}   \Vert_{H^{1}(0,1)}$
& $\Vert  p_{n}- p_{n - 1}  \Vert_{L^{2}(0,1)} $ & $\Vert  p_{n}- p_{n-1}  \Vert_{H^{1}(0,1)}$  \\
    \toprule
6 & $3^{6} + 1$ &  0.011432576436 &  0.050569870333 & 0.008280418446366  & 0.022150608375185 \\

7  & $3^{7} + 1$ & 0.006705537618 &  0.040243882138 & 0.004790279224858 & 0.014569849160095 \\

8  & $3^{8} + 1$ & 0.004020895404 & 0.033172226789 & 0.002725704395092 & 0.010008013980805 \\

9  & $3^{9} + 1$ & 0.000334472470 &  0.000634665497 & 0.001567106680915 & 0.007680602752328 \\
    \hline
\end{tabular}
\end{center}
\end{table*}
We observe convergence as predicted by the theoretical conclusions of \textit{Section} \ref{Sec A sigma finite Interface Microstructure}.
\begin{figure}[h]
\vspace{-1.3cm}
        \centering
        \begin{subfigure}[Solution $p^{4}$, Stage $n = 4$.]
                {\resizebox{5cm}{5cm}
                {\includegraphics{Ex1fnL4.pdf} } }
        \end{subfigure}
\qquad
        \begin{subfigure}[Solution $p^{9}$, Stage $n = 9$.]
                {\resizebox{5cm}{5cm}
{\includegraphics{Ex1fnL9.pdf} } }
        \end{subfigure} \\
        %
        %
          \begin{subfigure}[Derivative $\boldsymbol{\partial }  p^{4}$, Stage $n = 4$.]
                {\resizebox{5cm}{5cm}
                {\includegraphics{Ex1derL4.pdf} } }
        \end{subfigure}
        \qquad
        ~

          \begin{subfigure}[Derivative $\boldsymbol{\partial }  p^{9}$, Stage $n = 9$.]
                {\resizebox{5cm}{5cm}
{\includegraphics{Ex1derL9.pdf} } }
        \end{subfigure}
\caption{\textbf{Solutions \textit{Example} \ref{Ex Basic Example}}. Storage coefficient $\beta = 1$. The functions depicted in figures (a) and (b) are the solutions for the presence of microstructures $\mathsf{B} _{4}$ and $\mathsf{B} _{9}$ respectively, see \eqref{Def Partial Triadic Extremes Set}. Figures (c) and (d) are the respective derivatives. Forcing term $F = 0$ and interface forcing term $f$ defined in \eqref{Def Deterministic Interface Forcing Term}. The vertical lines in graphics of the derivatives are included for optical purposes only.}
\end{figure}
\end{example}
\begin{example}[Scaled Example]\label{Ex Scaled Example} In the second example we implement the model introduced in \textit{Section} \ref{Dec Fractal Scaling Modeling}, in order to illustrate the behavior of the solutions. We set $F = 0$, $f$ defined in \eqref{Def Deterministic Interface Forcing Term} above and the storage coefficient
\begin{equation}\label{Eq Example scaled storage coefficient}
\beta (\mathpzc{b} ) \defining   \sum_{n \, \in \, \boldsymbol{\mathbbm{N} } } \big(\frac{2}{3} \big)^{n}
\, \boldsymbol{\mathbbm{1} } _{\mathsf{B}_{n}  - \mathsf{B} _{n-1}}(\mathpzc{b} )\, .
\end{equation}
The exact solution can not be described exactly, due to its fractal nature. Therefore, we only present the Cauchy behavior of the sequence of solutions in the table below, together with the corresponding graphics. Again, convergence is observed as predicted by the theoretical results of \textit{Section} \ref{Dec Fractal Scaling Modeling}.
\begin{table*}[h!]
\centerline{\textit{Example} \ref{Ex Scaled Example} : Convergence Table, $\beta$ defined in \eqref{Eq Example scaled storage coefficient} .}\vspace{5pt}
\def1.4{1.4}
\begin{center}
\begin{tabular}{ c c c c }
    \hline
Stage $n$  & $\#$ Nodes  &
$\Vert  p_{n}- p_{n - 1}  \Vert_{L^{2}(0,1)} $ & $\Vert  p_{n}- p_{n-1}  \Vert_{H^{1}(0,1)}$  \\
    \toprule
6  & $3^{6} + 1$ & 0.002708076748262 &  0.005143218742685 \\

7  & $3^{7} + 1$ & 0.001344770703905 &  0.002552130987523 \\

8  & $3^{8} + 1$ & 0.000670086347099 &  0.001271522606439\\

9  & $3^{9} + 1$ & 0.000334472470474 &  0.000634665496467 \\
    \hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure}[h]
\vspace{-1.3cm}
        \centering
        \begin{subfigure}[Solution $p^{3}$, Stage $n = 3$.]
                {\resizebox{5cm}{5cm}
                {\includegraphics{Ex2fnL3.pdf} } }
        \end{subfigure}
\qquad
          \begin{subfigure}[Solution $p^{9}$, Stage $n = 9$.]
                {\resizebox{5cm}{5cm}
                {\includegraphics{Ex2fnL9.pdf} } }
        \end{subfigure}
 \\
        %
        %
                \begin{subfigure}[Derivative $\boldsymbol{\partial }  p^{3}$, Stage $n = 3$.]
                        {\resizebox{5cm}{5cm}
        {\includegraphics{Ex2derL3.pdf} } }
                \end{subfigure}
        \qquad
        ~

          \begin{subfigure}[Derivative $\boldsymbol{\partial }  p^{9}$, Stage $n = 9$.]
                {\resizebox{5cm}{5cm}
{\includegraphics{Ex2derL9.pdf} } }
        \end{subfigure}
\caption{\textbf{Solutions \textit{Example} \ref{Ex Scaled Example}}. Storage coefficient $\beta$ defined in \eqref{Eq Example scaled storage coefficient}. The functions depicted in figures (a) and (b) are the solutions for the presence of microstructures $\mathsf{B} _{4}$ and $\mathsf{B} _{9}$ respectively, see \eqref{Def Partial Triadic Extremes Set}. Figures (c) and (d) are the respective derivatives. Forcing term $F = 0$ and interface forcing term $f$ defined in \eqref{Def Deterministic Interface Forcing Term}. The vertical lines in graphics of the derivatives are included for optical purposes only.}
\end{figure}
\end{example}
\begin{example}[Random Behavior Example]\label{Ex Random Behavior Example}
This third and last example is a probabilistic variation of \textit{Example} \ref{Ex Basic Example}. We simply introduce uncertainty in the storage coefficient $\beta$ by considering it a random variable, uniformly distributed over an interval i.e.,
\begin{align}\label{Def Probabilistic Experimental Storage Coefficient}
& \beta: \mathsf{B} \rightarrow \Big[\frac{3}{4}\, , \frac{5}{4} \Big] \, , &
&  \beta\sim \text{uniformly} .
\end{align}
Due to the Law of Large Numbers, the average of $\beta$ is the constant function $\bar{\beta} = 1$. Therefore, due to the linearity of the differential equation, the ``averaged" function $\bar{p}$ of the solutions corresponding to the realizations is precisely the solution of \textit{Example} \ref{Ex Basic Example}. Below, we present the outcome of eight numerical experiments, each of them consisting in averaging the outcome of twenty random realizations; we denoted this average by $\bar{p}_{n}$. The experiments were executed for two different stages, namely $n = 4$ and $n = 8$, consequently finite versions of the storage coefficient $\beta$ \eqref{Def Probabilistic Experimental Storage Coefficient} were used, namely
\begin{align}\label{Def Finite Probabilistic Experimental Storage Coefficient}
& \beta_{i}: \mathsf{B} _{i}\rightarrow \Big[\frac{3}{4}\, , \frac{5}{4} \Big] \, , &
&  \beta_{i}\sim \text{uniformly} \, , &
&  i = 4, 8 .
\end{align}
\begin{table*}[h!]
\centerline{\textit{Example} \ref{Ex Random Behavior Example}: Experiments, Each Consisting of 20 Random Realizations.}\vspace{5pt}
\def1.4{1.4}
\begin{center}
 \begin{tabular}{ c c c c c c }
\hline
 & \multicolumn{2}{  c  }{Stage $n = 4$} & \multicolumn{2}{ c }{Stage $n = 8$ }\\
Experiment
& $\Vert  \, \bar{p}_{n}- \bar{p}  \, \Vert_{L^{2}(0,1)} $
& $\Vert  \, \bar{p}_{n}- \bar{p}  \,  \Vert_{H^{1}(0,1)} $
& $\Vert  \, \bar{p}_{n}- \bar{p}  \,  \Vert_{L^{2}(0,1)} $
&  $\Vert  \, \bar{p}_{n}- \bar{p}  \,  \Vert_{H^{1}(0,1)} $
\\
\toprule
1
& 0.000119184000335  & 0.001061092385691
& 0.000003969114134 &  0.000255622745433
\\
2
& 0.000212826056946 &  0.001524984778876
& 0.000011927845271 &  0.000729152212225
\\
3
& 0.000251807786501 &  0.002145242883204
& 0.000006707500555 &  0.000433471661184
\\
4
& 0.000118807313682  &  0.001071251072221
& 0.000006848283589 &  0.000399676826771
\\
5
& 0.000200559658088  &  0.001539679789493
& 0.000003353682741 &  0.000236079667248
\\
6
& 0.000274728964017  & 0.001805225518962
& 0.000017705174519  & 0.001055292921996
\\
7
& 0.000262099625343 &  0.001921561839120
& 0.000013849036017 &  0.000696257848189
\\
8
& 0.000338584909360 &  0.001974469520292
& 0.000004372910306 &  0.000289302820258
\\
\toprule
Average
& 0.000250408088105 &  0.001817080230904
& 0.000008011140143  & 0.000473741408661
\\
Variance
& $0.005804534\times 10^{-6}$  & $0.164983904 \times 10^{-6} $
&  $0.000027860\times 10^{-6}$  & $0.083787371 \times 10^{-6}$
\\
\hline
\end{tabular}
\end{center}
\end{table*}
In the table below the $L^{2}$ and $H^{1}$-norms for the difference of the partial averages $\bar{p}_{n}$ and the Ces\`aro average are displayed. Clearly convergence is observed in both cases and, due to the Law of Large Numbers, better behavior is observed at the stage $n = 8$ over the stage $n=4$, as expected. The average and variance are also presented at the bottom of the table, which shows better behavior for the more developed stage $n=8$ over the stage $n=4$ not only in terms of centrality, but also in terms of deviation.
\begin{figure}[h]
\vspace{-1.3cm}
        \centering
        \begin{subfigure}[Realization 1: Solution $p^{6}$.]
                {\resizebox{4.9cm}{4.9cm}
                {\includegraphics{RandFunc1.pdf} } }
        \end{subfigure}
\quad
        \begin{subfigure}[Realization 2: Solution $p^{6}$.]
                {\resizebox{4.9cm}{4.9cm}
{\includegraphics{RandFunc2.pdf} } }
        \end{subfigure}
\quad
        \begin{subfigure}[Realization 3: Solution $p^{6}$.]
                {\resizebox{4.9cm}{4.9cm}
{\includegraphics{RandFunc3.pdf} } }
        \end{subfigure} \\
        %
        %
          \begin{subfigure}[Realization 1: Derivative of the Solution $\boldsymbol{\partial }  p^{6}$.]
                {\resizebox{4.8cm}{4.8cm}
                {\includegraphics{RandDer1.pdf} } }
        \end{subfigure}
        \quad
        ~

          \begin{subfigure}[Realization 2: Derivative of the Solution $\boldsymbol{\partial }  p^{6}$.]
                {\resizebox{4.8cm}{4.8cm}
{\includegraphics{RandDer2.pdf} } }
        \end{subfigure}
        \quad
        ~

          \begin{subfigure}[Realization 3: Derivative of the Solution $\boldsymbol{\partial }  p^{6}$.]
                {\resizebox{4.8cm}{4.8cm}
{\includegraphics{RandDer3.pdf} } }
        \end{subfigure}
\caption{\textbf{Random Realizations \textit{Example} \ref{Ex Random Behavior Example}}. Storage coefficient $\beta$ defined in \eqref{Eq Example scaled storage coefficient}. For three random realizations, the functions depicted in figures (a), (b) and (c) are the solutions and figures (d), (e) and (f) the corresponding derivatives. The microstructure is $\mathsf{B} _{6}$, see \eqref{Def Partial Triadic Extremes Set}, forcing term $F = 0$ and interface forcing term $f$ defined in \eqref{Def Deterministic Interface Forcing Term}. The vertical lines in graphics of the derivatives are included for optical purposes only.}
\end{figure}
\end{example}
\subsection{Closing Observations}\label{Sec Closing Observations Numerical}
\begin{enumerate}[(i)]
\item
The authors tried to find, experimentally, a rate of convergence using the well-know estimate
\begin{align*}
& \alpha_{i} \sim \frac{\log \Vert  p^{ \, n+1}_{i}- p^{ \, n }_{i}  \Vert -
\log   \Vert  p^{ \, n}_{i} - p^{ \, n-1}_{i} \Vert }{
\log   \Vert  p^{ \, n}_{i} - p^{ \, n-1}_{i} \Vert
- \log \Vert  p^{ \, n-1}_{i}- p^{ \, n -2}_{i}  \Vert } \, ,&
& i = 1, 2.
\end{align*}
The sampling was made on the sequence of stages $3,4,5,6,7,8,9$. Experiments were run for \textit{Examples} \ref{Ex Basic Example} and \ref{Ex Scaled Example} however, in none of the cases, solid numerical evidence was detected that could suggest an order of convergence for the phenomenon.

\item Additional experiments for the Random Behavior \textit{Example} \ref{Ex Random Behavior Example} were executed. The probabilistic variations involved were

\begin{enumerate}[(a)]
\item
Experiments for the stages $n = 3, 5, 6, 7$ and $9$.

\item Experiments with different number of realizations, namely $20$, $60$ and $80$, depending on the computational time demanded.

\item Using a storage coefficient $\beta: \mathsf{B} \rightarrow I$, with $I$ different intervals and $\beta$ with uniform distribution normal distribution.

\item Probabilistic variations of the forcing terms.

\item Combinations of one or more of the previous factors.

\item Execution of the aforementioned probabilistic variations adjusted to the scaled \textit{Example} \ref{Ex Scaled Example}.

\end{enumerate}
In all the cases, convergence behavior was observed as expected. Naturally, the quality of convergence deteriorates depending on the deviation of the distributions and the combination of uncertainty factors introduced in the experiment.

\item All the previously mentioned scenarios were also executed in the same code for the case of homogeneous Dirichlet boundary conditions on both ends i.e., $p_{n} (0) = p_{n} (1) = 0$. As expected convergence behavior is observed which is comparable with the corresponding analogous version setting the boundary conditions $p_{n} (0) = 0$, $\boldsymbol{\partial }  p_{n} (1) = 0$, used along the analysis of this paper.

\end{enumerate}
\section{Conclusions and Final Discussion}
The present work yields several accomplishments as well as limitations listed below.
\begin{enumerate}[(i)]
\item The unscaled storage model presented in \textit{Section} \ref{Sec A sigma finite Interface Microstructure} is in general not adequate, as it excludes most of the important cases of fractals.

\item The scaled storage model presented in \textit{Section} \ref{Dec Fractal Scaling Modeling} is suitable for an important number of fractal microstructures. However, the asymptotic variational model \eqref{Pblm Variational Microstructure Good Coefficients} is equivalent to a pointwise strong model \eqref{Pblm Strong Microstructure Good Coefficients} only if the closure of the microstructure $\cl(\mathsf{B} )$ is negligible i.e., if it has null Lebesgue measure. Such hypothesis excludes important cases e.g. the family of ``fat" Cantor sets, see \cite{Falconer} in 1-D or the ``fat" versions of the classic fractal structures in 2-D or 3-D, where only the ``averaged normal flux" \textit{Statement} \eqref{Eq strong limit interface accumulation points extra hypothesis} can be concluded for the accumulation points of the microstructure $\mathsf{B} '$.

\item In order to overcome the deficiency previously mentioned a first approach would be to take the traditional treatment of solving strong forms in fractal domains (e.g. \cite{Strichartz3, Strichartz1}) and then try to ``blend" it with the point of view presented here. Such analysis is to be pursued in future research.

\item The fractal microstructures addressed in this work are self-similar. This requirement is important only for the second model (\textit{Section} \ref{Dec Fractal Scaling Modeling}) because it scales the storage \eqref{Eq scaled storage coefficient} according to the geometric detail of the structure at every level. In particular, we need an accurate idea of the growth rate of the microstructure $\mathsf{B} $ from one level to the next, in order to scale the storage adequately.

\item The self-similarity requirement for the microstructure $\mathsf{B} $ in the introduction of the scaled model excludes the important family of the self-affine fractals; this type of microstructures is a topic for future work. On the other hand, the unscaled model of \textit{Section} \ref{Sec A sigma finite Interface Microstructure} does not require such detailed knowledge of the microstructure precisely because it avoids scaling, although it is likely to be unsuited for the analysis of self-affine microstructures it may be a good starting point in order to detect the needs of the modeling for this case.

\item It is important to observe the relevance of self-similarity versus the fractal dimension in the scaled model. While the self-similarity of the microstructure is the corner stone of the scaling \eqref{Eq scaled storage coefficient} (hence it can not be given up) the model is more flexible with respect to the fractal dimension of $\mathsf{B} $ as long as it is not the same of the ``host" domain $\Omega$.

\item We also stress that the input needed by the present result, in terms of geometric information on the fractal structure, does not have to be as detailed as in the strong forms PDE analysis on fractals.

\item The random experiments presented in \textit{Example} \ref{Ex Random Behavior Example}, as well as those only mentioned in \textit{Subsection} \ref{Sec Closing Observations Numerical}, furnish solid numerical evidence of good behavior for probabilistic versions of the unscaled and the scaled models respectively. Additionally, it is important to handle certain level of uncertainty because, assuming to have a deterministic description of the fractal microstructure is a too strong hypothesis to be applicable in realistic scenarios. In the Authors' opinion, this is justification enough to pursue rigorous analysis of these problems to be addressed in future work.

\item In \textit{Example} \ref{Ex Random Behavior Example} uncertainty was introduced in the storage coefficient $\beta$, or the forcing term $f$, however the geometry of the microstructure was never randomized. In several works (e.g. \cite{GrafS, HutchinsonRuschendorf}) the self-similarity is replaced by the concept of statistical self-similarity in the sense that scaling of small parts have the same statistical distribution as the whole set. Clearly, this random property is consistent with the scaling of $\beta$ \eqref{Eq scaled storage coefficient} in \textit{Definition} \ref{Def scaled storage coefficient}; consequently, the statistical self-similarity is a future line of research for handling geometric uncertainty of the fractal microstructure $\mathsf{B} $. In particular, the fractal percolation microstructures, are of special interest for real world applications, see \cite{ChayesDurret, DekkingMeester}.

\item The execution of all the numerical experiments shows that the code becomes unstable beyond the 9th stage of the Cantor set construction. This suggests that in order to overcome these issues, an adaptation of the FEM method has to be developed, targeted to the microstructure of interest. This aspect is to be analyzed in future research.

\item
The study of fractal microstructures in 2-D and 3-D are necessary for practical applications and in higher dimensions for theoretical purposes. Given that passing from one dimension to two or more dimensions increases significantly the level of complexity in the microstructure and in the equation, considerable challenges are to be expected in this future line of research.

\item Another important analysis to be developed is the study of the models both, scaled and unscaled, in the mixed-mixed variational formulation introduced in \cite{MoralesShow2}. On one hand, this approach allows great flexibility for the underlying spaces of velocity and pressure which can constitute an advantage with respect to the treatment presented here. On the other hand, the mixed-mixed variational formulation allows modeling fluid exchange conditions across the interface of greater generality than  \eqref{Eq strong n-stage interface}, used in the present work; this fact can contribute significantly to the development of the field.

\end{enumerate}

\section*{Acknowledgements}
The authors wish to acknowledge Universidad Nacional de Colombia, Sede Medell\'in for its support in this work through the project HERMES 27798. The authors also wish to thank Professor Ma\l{}gorzata Peszy\'nska from Oregon State University, for authorizing the use of code \textbf{fem1d.m} \cite{PeszynskaFEM} in the implementation of the numerical experiments presented in \textit{Section} \ref{Sec Numerical Experiments}. It is a tool of remarkable quality, efficiency and versatility which has contributed significantly to this work.

redpajama_set_name: RedPajamaArXiv


Location: Morrisonville, New York, U.S.A.
The Clinton County Sanitary Landfill covers almost 70 acres and receives approximately 175,000 metric tons of waste each year. To reduce the greenhouse gas pollution from this waste, New England Waste Services of New York (NEWS), the landfill operators, voluntarily built a landfill methane gas collection and destruction system.
The system includes vertical and horizontal gas wells, collection laterals, a blower, and a flare to combust the gas. In April 2005, NEWS began voluntarily flaring the landfill gas. In 2008, NEWS constructed a 4.8 MW landfill gas-to-energy power plant that combusts all of the gas in reciprocating engines to generate power for the local community.
This power project became operational in October 2008, and it is expected to eliminate methane emissions equivalent to 100,000 metric tons of CO2/year. This action is comparable to the climate benefits of reducing more than 223,000,000 auto miles per year.
This project is a collaborative effort between NEWS, which owns and operates the landfill's waste-to-energy generation; Clinton County, which owns the landfill; and NativeEnergy, which has contracted to purchase the verified carbon offsets associated with certain methane emission reductions from the project.
Municipal solid waste landfills are the second largest source of human-related methane emissions in the United States, accounting for approximately 23 percent of these emissions in 2007. This project, in addition to producing renewable electricity, traps and destroys greenhouse gases otherwise emitted from the landfill.
Additionally, the project provides good economic use of otherwise unusable land by transforming what was once considered waste into consumable energy. Revenues from the project's sales are shared between Clinton County and the project's operator, NEWS.
The Clinton County Landfill project will be listed with and verified to the Climate Action Reserve standard, and its VERs will be retired in the Climate Reserve Registry.
The Clinton County Landfill project demonstrates financial additionality according to UNFCCC definitions. The opportunity to realize revenues from the offset/REC sales was necessary to Casella's determination and ability to implement the project. This project has been validated to the landfill gas protocol of the Climate Action Reserve, and its CRT production will be third-party verified annually.