text:
\section{Introduction}

Many systems that result in steady-state oscillations can be modeled with the Helmholtz equation, but of particular interest are acoustic waves and transverse electric or transverse magnetic electromagnetic waves in inhomogeneous media.  In each of these situations, the equation of interest can be expressed as
\begin{equation}\label{acoustic}-\nabla\cdot\rho^{-1}\nabla P-\frac{\omega^2}{\kappa}P=0\end{equation}
for appropriate choices of the complex-valued, spatially dependent material parameters $\rho$ and $\kappa,$ where $\omega > 0$ is the frequency.  The classical methods of deriving a weak form for this equation (with Dirichlet boundary conditions, for example) result in the variational equation
\[\int_\Gamma \left[\rho^{-1}\nabla P\cdot\nabla \overline u-\frac{\omega^2}{\kappa}P\overline u\right] dx=0, \ \ \ \forall u\in H_0^1(\Gamma),\]
which corresponds to a stationary principle, but not a minimization principle.  In \cite{Graeme}, Milton, Seppecher, and Bouchitt\'{e} expand upon the work of Cherkaev and Gibiansky \cite{Cherk} for the conductivity equation to derive variational principles for (\ref{acoustic}) (as a special case of the more general equations of elasticity and electromagnetism) that are true minimization principles,
provided the media are lossy.  The minimization functional corresponds physically to dissipated energy in the system, and is valid
even for arbitrarily small coefficients of loss.
While the framework presented in \cite{Graeme} results in nonstandard boundary conditions, Milton and Willis extend the principles to handle the classical Dirichlet and Neumann boundary conditions in \cite{Willis}.

In this paper we apply the finite element method to these minimization principles and thereby develop a numerical algorithm for solving (\ref{acoustic}) that can take advantage of the many efficient methods available for solving a symmetric, positive-definite system of linear equations.  The outline of the paper is as follows.  Sections~\ref{model} and \ref{bcs} review the general variational formulation and boundary conditions introduced by Milton, Seppecher, Bouchitt\'{e}, and Willis.  In Section~\ref{error_section}, we derive an error bound on certain finite element discretizations of the variational principle.  In Sections~\ref{el_section} and \ref{implement}, we describe a straightforward implementation of the finite element method on a square domain, with Dirichlet boundary conditions.
In Section~\ref{conditioning_section}, we suggest a preconditioner for solving the resulting symmetric positive definite linear system via the preconditioned conjugate gradient method, and find conditions on the material coefficients under which we expect the best conditioning.  Section~\ref{numerical_section} describes the results of some simple numerical experiments, and illustrates numerical convergence consistent with the error bounds from Section~\ref{error_section}.  Finally, in Section~\ref{robin}, we extend the method to handle Robin boundary conditions, and present some associated numerical examples.

\section{Variational Formulation}

Our model problem is
\begin{equation}\label{model}
\left\{\begin{array}{ll}
\displaystyle -\nabla\cdot\rho^{-1}\nabla P-\frac{\omega^2}{\kappa}P=0 & \mbox{in }\Gamma \\
P=f & \mbox{on }\partial \Gamma \\
\end{array}\right.
\end{equation}
where $\Gamma$ is an open, bounded subset of $\mathbb{R}^d$ ($d=2$ or $3$) with smooth boundary.  For acoustic waves, $\rho$ is the density, $\kappa$ is the bulk modulus, $\omega$ is the frequency, and $P$ is the pressure.  Here $\rho,\ \kappa,$ and $P$ are all complex.  In this section, we focus on Dirichlet boundary conditions for simplicity; Neumann conditions can be handled similarly.  In \cite{Graeme}, it is shown in detail how this and other problems can be formulated as a minimization.  What follows is a brief outline of the general framework.

Let $\mathcal{F}(x)$ and $\mathcal{G}(x)$ be complex-valued fields of the form
\[\mathcal{F}=\left(\begin{array}{c}
F \\
f \\
\end{array}\right)\ \mathcal{G}=\left(\begin{array}{c}
G \\
g \\
\end{array}\right),\]
where $F,\ G:\Gamma\rightarrow\mathbb{C}^d$ and $f,\ g:\Gamma\rightarrow\mathbb{C}$.  Suppose there exists a complex-valued potential $u$ such that
\[\mathcal{F}=\sqcap u:=\left(\begin{array}{c}
\nabla u \\
u \\
\end{array}\right)\]
and that $\mathcal{G}$ satisfies
\[h+\sqcup\mathcal{G}=0,\]
where $\sqcup\mathcal{G}:=-\nabla\cdot G+g$ and $h$ is a source term.  Suppose also that $\mathcal{F}$ and $\mathcal{G}$ satisfy the constitutive relation
\begin{equation}\label{constitutive}\mathcal{G}(x)=Z(x)\mathcal{F}(x)\end{equation}
where $Z$ has the form
\[Z=\left(\begin{array}{cc}
L & K \\
K^{T} & M \\
\end{array}\right).\]
Then the constitutive relation along with the differential constraints imply
\begin{equation}\label{one} h+\sqcup(Z\sqcap u)=0\mbox{ or }\nabla\cdot(L\nabla u+Ku)=h+K^{T}\nabla u+Mu\end{equation}
Let $n$ be the unit outward normal on $\partial\Gamma$.  It is shown in \cite{Graeme} that if we are given $u_0'$ and $G_0'\cdot n$ and we specify
\begin{equation}\label{two}u'=u_0'\mbox{ and } G'\cdot n=G_0'\cdot n\mbox{ on }\partial\Gamma\end{equation}
(herein $'$ denotes the real part of a complex quantity and $''$ the imaginary part), then the solution to (\ref{one}) satisfying the boundary conditions (\ref{two}) is a minimizer of the functional
\[Y(u',G')=\int_\Gamma\left[\left(\begin{array}{c}
\sqcap u' \\
-\mathcal{G}' \\
\end{array}\right)\cdot\mathcal{L}\left(\begin{array}{c}
\sqcap u' \\
-\mathcal{G}' \\
\end{array}\right)+2h''u'\right] dx,\]
where
\[\mathcal{G}'=\left(\begin{array}{c}
G' \\
\nabla\cdot G'-h \\
\end{array}\right)\mbox{ and }\mathcal{L}=\left(\begin{array}{cc}
Z''+Z'(Z'')^{-1}Z' & Z'(Z'')^{-1} \\
(Z'')^{-1}Z' & (Z'')^{-1} \\
\end{array}\right),\]
provided that $\mathcal{L}$ is positive definite.  An inspection of the constitutive relation shows that $\mathcal{L}$ is positive definite as long as $Z''$ is.  Explicitly, following \cite{Graeme} we see that if we let $\mathcal{F}'$ and $\mathcal{G}'$ be arbitrary, and define $\mathcal{G}''$ and $\mathcal{F}''$ by
\[\left(\begin{array}{c}
\mathcal{G}'' \\
\mathcal{F}'' \\
\end{array}\right)=\mathcal{L}\left(\begin{array}{c}
\mathcal{F}' \\
-\mathcal{G}' \\
\end{array}\right),\]
which is equivalent to
 \[ \begin{array}{l}
\mathcal{G}'=Z'\mathcal{F}'-Z''\mathcal{F}'' \\
\mathcal{G}''=Z'\mathcal{F}''+Z''\mathcal{F}' \\
\end{array},\]
then
\[\left(\begin{array}{c}
\mathcal{F}' \\
-\mathcal{G}' \\
\end{array}\right)\cdot\mathcal{L}\left(\begin{array}{c}
\mathcal{F}' \\
-\mathcal{G}' \\
\end{array}\right)=\mathcal{F}'\cdot\mathcal{G}''-\mathcal{F}''\cdot\mathcal{G}'\]
\[=\mathcal{F}'\cdot(Z'\mathcal{F}''+Z''\mathcal{F}')-\mathcal{F}''\cdot(Z'\mathcal{F}'-Z''\mathcal{F}'' )\]
\[=\mathcal{F}'\cdot Z''\mathcal{F}'+\mathcal{F}''\cdot Z''\mathcal{F}''.\]
Therefore, $\mathcal{L}$ is positive definite as long as $Z''$ is.

\section{Boundary Conditions}\label{bcs}

In addition to the conditions
\[u'=u_0'\mbox{ and }G'\cdot n=G_0'\cdot n\mbox{ on }\partial\Gamma\]
we can also solve the problem for $u'$ and $G'$ with the boundary conditions
\begin{equation}\label{three}u''=u_0''\mbox{ and }G''\cdot n=G_0''\cdot n\mbox{ on }\partial\Gamma,\end{equation}
\[u'=u_0'\mbox{ and }u''=u_0''\mbox{ on }\partial\Gamma,\]
\[\mbox{or }G'\cdot n=G_0'\cdot n\mbox{ and }G''\cdot n=G_0''\cdot n\mbox{ on }\partial\Gamma.\]
The correct variational principles for the last two sets of boundary conditions can be deduced from the formulations for the first two.  The second boundary condition above is a condition on the dual (imaginary) variables $u''$ and $G''$, and therefore it may be enforced through boundary integrals, as follows.

For simplicity, suppose $h=0$.  Let $s\in H^1(\Gamma)$ and $T\in H(\mbox{div},\Gamma)$.  If $u$ and $G$ are such that the differential constraints and constitutive relation are satisfied, then multiplying by $s$ and integrating, we get
\[0=\int_\Gamma(\sqcup \mathcal{G})''s\ dx=\int_\Gamma\left[(-\nabla\cdot G''+g'')s\right]\ dx=\int_\Gamma\left[(-\nabla\cdot G''+g'')s-T\cdot(\nabla u''-\nabla u'')\right]\ dx.\]
Integrating by parts, we find
\[\int_\Gamma\left[G''\cdot\nabla s+g''s-T\cdot\nabla u''-u''\nabla\cdot T\right]\ dx=\int_{\partial\Gamma}\left[sG''\cdot n-u''T\cdot n\right]\ dS.\]
Let $\mathcal{T}=(T,\nabla\cdot T)^T$.  The left-hand side above can be re-written as
\[\int_\Gamma\left(\begin{array}{c}
\mathcal{G}'' \\
\mathcal{F}'' \\
\end{array}\right)\cdot\left(\begin{array}{c}
\sqcap s \\
\mathcal{T} \\
\end{array}\right)\ dx=\int_\Gamma\left(\begin{array}{c}
\sqcap u' \\
\mathcal{G}' \\
\end{array}\right)\cdot\mathcal{L}\left(\begin{array}{c}
\sqcap s \\
\mathcal{T} \\
\end{array}\right)\ dx.\]
The corresponding functional for the boundary condition
\[u''=u_0''\mbox{ and }G''\cdot n=G_0''\cdot n\mbox{ on }\partial\Gamma\]
is then
\[\tilde Y(u',G')=\int_\Gamma\left(\begin{array}{c}
\sqcap u' \\
\mathcal{G}' \\
\end{array}\right)\cdot\mathcal{L}\left(\begin{array}{c}
\sqcap u' \\
\mathcal{G}' \\
\end{array}\right)\ dx+2\int_{\partial\Gamma}\left[u_0''G'\cdot n-u'G_0''\cdot n\right]\ dS.\]
To solve the PDE with the Dirichlet boundary conditions we minimize the functional
\[\hat Y(u',G')=Y(u'+u_0',G')+2\int_{\partial\Gamma}u_0''G'\cdot n\ dS\]
over $u'\in H_0^1(\Gamma)$ and $G'\in H(\mbox{div},\Gamma)$.  To solve the PDE with the Neumann boundary conditions we minimize the functional
\[\check Y(u',G')=Y(u',G'+G_0')-2\int_{\partial\Gamma}u'G_0''\cdot n\ dS\]
over $u'\in H^1(\Gamma)$ and $G'\in H_0(\mbox{div},\Gamma)=\{v\in H(\mbox{div},\Gamma):\langle v\cdot n,w\rangle=0\ \forall\ w\in H_0^1(\Gamma)\}$ (see \cite{Brezzi-Fortin}).

\section{Error Bound}\label{error_section}

In this section we give a bound on the error incurred by solving any of the minimization problems above over a finite dimensional subspace of the specified Sobolev spaces.  We will give a more detailed account of exactly what the finite dimensional space looks like later on; in this section all that will matter is the highest degree of polynomials that the finite dimensional space contains.  We will use the Bramble-Hilbert lemma to give a bound on the error.

Here we will drop the primes used to denote real and imaginary parts.  Note that what follows applies to any of the boundary value problems discussed previously, since the bounds depend only on the corresponding bilinear form.  Throughout this section, $C$ is a constant independent of the solution $(P,v)$ and the grid spacing $h$.

\subsection{Bilinear Form}
Define the bilinear form $B$ by
\begin{equation}\label{bilinear}B(P,v;s,T)=\int_\Gamma\left(\begin{array}{c}
\mathcal{F} \\
-\mathcal{G} \\
\end{array}\right)\cdot\mathcal{L}\left(\begin{array}{c}
\mathcal{S} \\
-\mathcal{T} \\
\end{array}\right)\ dx,\end{equation}
Where, as before, $\mathcal{F}=\sqcap u$, $\mathcal{G}=(G,\nabla\cdot G)^T$, and $\mathcal{S}$ and $\mathcal{T}$ are generated from test function $s\in H^1(\Gamma)$ and $T\in H(\mbox{div},\Gamma)$ in the same fashion.  Assume that there exist constants $\gamma_1,\gamma_2>0$ such that $\mathcal{L}>\gamma_2 I$ and that $[\mathcal{L}(x)]_{ij}\le \gamma_1$ for a.e. $x\in \Gamma$.  Let $V=H_0^1(\Gamma)\times H(\mbox{div},\Gamma)$, endowed with the norm
\[\|(u,G)\|_V=(\|u\|_{H^1(\Gamma)}^2+\|G\|_{H(\mbox{div},\Gamma)}^2)^{\frac{1}{2}}.\]
Then it follows immediately from (\ref{bilinear}) that
\begin{equation}\label{continuous}B(u,G;s,T)\le C\gamma_1\|(u,G)\|_V\|(s,T)\|_V\end{equation}
and
\begin{equation}\label{coercive}B(u,G;u,G)\ge\gamma_2\|(u,G)\|_V^2.\end{equation}

\subsection{Minimization Inequality}
Define an energy by
\[f(s,T)=\frac{1}{2}B(s,T;s,T)-F(s,T),\]
where $F:H^1(\Gamma)\times H(\mbox{div},\Gamma)\rightarrow\mathbb{R}$ (in practice, $F$ is usually composed of terms resulting from an inhomogeneous term and enforcement of the desired boundary conditions). If $(u,G)$ is the minimizer of the energy, then this pair must satisfy the Euler-Lagrange equation
\[B(u,G;s,T)=F(s,T)\ \ \ \forall s\in H_0^1(\Gamma),\ \forall\ T\in H(\mbox{div},\Gamma),\]
so that
\[f(s,T)=f(u ,G)+\frac{1}{2}B(u-s, G-T;u-s,G-T)\ \ \ \forall\ s\in H_0^1(\Gamma)\ \forall\ T\in H(\mbox{div},\Gamma).\]
Consider a finite dimensional subspace $V_N=V_{N1}\times V_{N2}$ of $V$, where $V_{N1}$ is a finite dimensional subspace of $H^1(\Gamma)$ and $V_{N2}$ is a finite dimensional subspace of $H(\mbox{div},\Gamma)$.  If $(u_N,G_N)$ is such that
\[f(u_N,G_N)=\min_{(s,T)\in V_N} f(s,T),\]
then
\[\left[B(u-u_N,G-G_N;u-u_N,G-G_N)\right]^{\frac{1}{2}}=\min_{(s,T)\in V_N}\left[B(u-s,G-T;u-s,G-T)\right]^{\frac{1}{2}}.\]
Inequalities (\ref{continuous}) and (\ref{coercive}) imply that
\[\sqrt{\gamma_2}\|(s,T)\|_V\le\sqrt{B(s,T;s,T)}\le C\sqrt{\gamma_1}\|(s,T)\|_V\ \ \ \forall\ (s,T)\in V,\]
so we have
\begin{equation}\label{min_test}\sqrt{\gamma_2}\|(u,G)-(u_N,G_N)\|_V\le\min_{(s,T)\in V_N} C \sqrt{\gamma_1}\|(u,G)-(s,T)\|_V.\end{equation}
Let $F_1$ be the orthogonal projection from $H^1(\Gamma)$ onto $V_{N1}$.  Since $F_1$ is an orthogonal projection, it has $\|F_1\|_{B(H^1(\Gamma),H^1(\Gamma))}=1$, where $B(H^1(\Gamma),H^1(\Gamma))$ is the set of bounded linear functions from $H^1(\Gamma)$ to $H^1(\Gamma)$.  Also, define an operator $F_2:H(\mbox{div},\Gamma)\rightarrow V_{N2}$ by the solution of the variational inequality
\[\left<F_2G,Q-F_2G\right>_{L^2(\Gamma,\mathbb{R}^d)}\ge\left<G,Q-F_2G\right>_{L^2(\Gamma,\mathbb{R}^d)}\ \forall\ Q\in E_G,\]
over the set $E_G=\{v\in V_{N2}:\|\nabla\cdot v\|_{L^2(\Gamma)}\le \|\nabla\cdot G\|_{L^2(\Gamma)}\}$, which is a closed, convex subset of $L^2(\Gamma,\mathbb{R}^d)$.  We then have
\[\|F_2 G\|_{L^2(\Gamma,\mathbb{R}^d)}^2=\left<F_2 G,F_2 G\right>_{L^2(\Gamma,\mathbb{R}^d)}\le\left<G,F_2 G\right>_{L^2(\Gamma,\mathbb{R}^d)}\le\|G\|_{L^2(\Gamma,\mathbb{R}^d)}\|F_2 G\|_{L^2(\Gamma,\mathbb{R}^d)}.\]
If we take $s=F_1u$ and $T=F_2 G$ in (\ref{min_test}), then we have
\begin{equation}\label{minimization}\|(u,G)-(u_N,G_N)\|_V\le C\|(u-F_1u,G-F_2G)\|_V.\end{equation}

\subsubsection{Seminorm bounds}

We will discretize the domain $\Gamma$ by by subdividing it into smaller regions, each of which can be seen as a suitable shifting and scaling of a reference element.  More precisely, if $\hat e$ is our reference element, there exist affine changes of variables $F_l(x)=Bx+x_l$ such that $F_l(\hat e)=e_l$, where $e_l$ is the $l$th element (subdivision) in the finite element decomposition of $\Gamma$.  In the case of rectangular elements in $\mathbb{R}^d$, for example, we can take $\hat e=(0,1)^d$, and then we have $B=hI_d$.  In this section a hat will denote the corresponding function defined over the reference element.

Let
\begin{equation}\label{seminorm}[u,w]_s=\sum_{|\alpha|=s}\int_{\hat e}D^{\alpha}u\cdot D^{\alpha}w\ dx\ \mbox{ and }\ |w|_s^2=[w,w]_s, \\
\end{equation}
where for vector functions we define
\[D^{\alpha}w=\left(\begin{array}{c}
D^{\alpha}w_1 \\
D^{\alpha}w_2 \\
\vdots \\
D^{\alpha}w_d \\
\end{array}\right).\]
 From \cite{Brezzi-Fortin} we get the inequalities
\begin{equation}\label{inequalities}\begin{array}{ll}
c^{-1}h^{s-\frac{d}{2}}|w|_{s,e_l}\le|\hat w|_s\le ch^{s-\frac{d}{2}}|w|_{s,e_l} \\
h^{s+\frac{d}{2}-1}|q|_{s,e_l}\le|\hat q|_s\le h^{s+\frac{d}{2}-1}|q|_{s,e_l} \\
h^{s+\frac{d}{2}}|\nabla\cdot q|_{s,e_l}\le |\nabla\cdot\hat q|_s\le h^{s+\frac{d}{2}}|\nabla\cdot q|_{s,e_l} \\
\end{array}\end{equation}
for $B=h$, scalar functions $w$, and vector functions $q$, where $w=\hat w\circ F^{-1}$ and $q=\hat q\circ F^{-1}$ and $|\cdot|_{s,e_l}$ denotes (\ref{seminorm}) with $e_l$ in place of $\hat e$.

We now recall the following lemma from \cite{Axelsson-Barker}, which will be used in what follows.
\begin{lma}[Bramble-Hilbert Lemma] For some region $\Omega\subset\mathbb{R}^2$ and some integer $k\ge -1$, let there be given a bounded linear functional
\[f:H^{k+1}(\Omega)\rightarrow\mathbb{R},\]
satisfying $|f(u)|\le\delta\|u\|_{H^{k+1}(\Omega)}$ for all $u\in H^{k+1}(\Omega)$ for some $\delta$ independent of $u$.  Suppose that $f(u)=0$ for all $u\in P_k(\bar\Omega)$.  Then there exists a constant $C$, dependent only on $\Omega$ such that
\[|f(u)|\le C\delta|u|_{k+1},\ \ \ u\in H^{k+1}(\Omega).\]
\end{lma}

Let us suppose that $\hat P\in H^{k+1}(\hat e)$ and $\hat v\in H^{j}(\mbox{div},\hat e)=\{q\in H^j(\hat e,\mathbb{R}^d):\nabla\cdot q\in H^j(\hat e)\}$.  For fixed elements $w\in H^s(\hat e)$ and $Q \in H^s(\mbox{div},\hat e)$ define the functionals
\[f_1(u)=[u-F_1u,w]_s,\ f_2(G)=[G-F_2G,Q]_0,\ f_3(\nabla\cdot G)=[\nabla\cdot G-\nabla\cdot F_2G,\nabla\cdot Q]_0,\]
where $s=0$ or $s=1$.  Then, since
\[|f_1(u)|\le |u-F_1u|_s|w|_s\le (|u|_s+|F_1u|_s)|w|_s\le(\|u\|_{H^1(\Gamma)}+\|F_1u\|_{H^1(\Gamma)})|w|_s\]
\[\le 2\|u\|_{H^1(\Gamma)}|w|_s\le2\|u\|_{H^{k+1}(\Gamma)}|w|_s,\]
\[ |f_2(G)|\le |G-F_2G|_0|Q|_0\le (|G|_0+|F_2G|_0)|Q|_0=(\|G\|_{L^2(\Gamma,\mathbb{R}^d)}+\|F_2G\|_{L^2(\Gamma,\mathbb{R}^d)})|Q|_0\]
\[\le2\|G\|_{L^2(\Gamma),\mathbb{R}^d)}|Q|_0\le\|G\|_{H^j(\Gamma,\mathbb{R}^d)}|Q|_0,\]
\[ |f_3(\nabla\cdot\ G)|\le |\nabla\cdot G-\nabla\cdot F_2G|_0 |\nabla \cdot Q|_0\le (|\nabla\cdot G|_0+|\nabla\cdot F_2G|_0)|\nabla\cdot Q|_0\]
\[=(\|\nabla\cdot G\|_{L^2(\Gamma)}+\|\nabla\cdot F_2 G\|_{L^2(\Gamma)})|\nabla\cdot Q|_0\le 2\|\nabla\cdot G\|_{L^2(\Gamma)}|\nabla\cdot Q|_0\le 2\|\nabla\cdot G\|_{H^j(\Gamma)}|\nabla\cdot Q|_0,\]
and $F_1u=u$ for polynomials in $V_{N1}$ and $F_2G=G$ for vectors of polynomials from $V_{N2}$, we can apply the Bramble-Hilbert lemma to find that there exists a constant such that
\[|f_1(\hat u)|\le C |w|_s |\hat u|_{k+1},\ |f_2(\hat G)|\le C|Q|_0 |\hat G|_{j},\ |f_3(\nabla \cdot \hat G)| \le C |\nabla \cdot Q|_0|\nabla\cdot\hat G|_{j},\]
as long as $k$ and $j$ are small enough so that all polynomials of degree less than or equal to $k$  are contained in the span of the basis functions representing $\hat u$ and all polynomials of degree less than or equal to $j$ are contained in the span of the basis functions representing $\hat G$.  By choosing $w=\hat u-F_1\hat u$ and $Q=\hat G-F_2\hat G$, we find that
\[|\hat u-F_1\hat u|_s \le C |\hat u|_{k+1},\ |\hat G-F_2\hat G|_0\le C |\hat G|_{j},\ |\nabla\cdot\hat G-\nabla\cdot F_2\hat G|_0\le C|\nabla\cdot \hat G|_{j}.\]

Employing (\ref{inequalities}), we find that for $h\le 1$,
\[|u-F_1u|_{s,e_l}\le C h^{\frac{d}{2}-s}|\hat u-F_1\hat u|_s\le C h^{\frac{d}{2}-s}|\hat u|_{k+1}\le C h^{k-s+1}|u|_{k+1,e_l},\]
\[|G-F_2G|_{0,e_l}\le h^{1-\frac{d}{2}}|\hat G-F_2\hat G|_0\le h^{1-\frac{d}{2}}C|\hat G|_{j}\le Ch^{j}|G|_{j,e_l},\]
\[|\nabla\cdot G-\nabla\cdot F_2G|_{0,e_l}\le h^{-\frac{d}{2}}|\nabla \cdot \hat G-\nabla\cdot F_2\hat G|_{0} \le h^{-\frac{d}{2}}C|\nabla\cdot\hat G|_{j}\le C h^{j}|\nabla \cdot G|_{j,e_l}.\]
Returning to inequality (\ref{minimization}), we have
\[\|(u,G)-(u_N,G_N)\|_V^2\le C\|(u,G)-(F_1u,F_2G)\|_V^2\]
\[=C \sum_l\left[|u-F_1 u|_{0,e_l}^2+|u-F_1u|_{1,e_l}^2+|G-F_2 G|_{0,e_l}^2+|\nabla\cdot v-\nabla\cdot F_2 G|_{0,e_l}^2\right]\]
\[\le C \sum_l \left[h^{2k+2}|u|_{k+1,e_l}^2+h^{2k}|u|_{k+1,e_l}^2+h^{2j}|G|_{j,e_l}^2+h^{2j}|\nabla \cdot G|_{j,e_l}^2\right] \]
\[ \le C (h^{2k} |u|_{k+1,\Gamma}^2+h^{2j}(|G|_{j,\Gamma}^2+|\nabla\cdot G|_{j,\Gamma}^2)).\]

Let $P_k(\bar \Gamma)$ denote all polynomials of degree less than or equal to $k$ on $\bar \Gamma$.  We have now proved
\begin{thm}\label{result}If the solution $(u,G)\in H^{k+1}(\Gamma)\times H^{j+1}(\mbox{div},\Gamma)$ and the finite element subspace used in the numerical method contains $P_k(\bar \Gamma) \times P_j(\bar \Gamma)) \times P_j(\bar \Gamma)$, then there is a constant $C$ such that the error satisfies
\[\|(u,G)-(u_N,G_N)\|_V^2\le C (h^{2k} |u|_{k+1,\Gamma}^2+h^{2j}(|G|_{j,\Gamma}^2+|\nabla\cdot G|_{j,\Gamma}^2)),\]
where $h\le 1$ is the grid spacing.
\end{thm}

\subsection{Regularity}
In order for the error bound to be meaningful, we must have $k,j \ge 1$ in Theorem~\ref{result}, which means that at least
\[u\in H^2(\Gamma)\mbox{ and }G\in H^1(\mbox{div},\Gamma).\]
In the notation of the acoustic equation, if $\rho^{-1}$ is positive definite, bounded, and $C^1$, then classical elliptic regularity theory such as in \cite{Evans} guarantees that $P'\in H^2(\Gamma)$.  Also since
\[v=-\frac{i}{\omega}\rho^{-1}\nabla P,\]
we have that $v\in (H^1(\Gamma))^2$, and multiplying the acoustic equation through by $-1/\omega$ tells us that
\[\nabla\cdot v=\frac{i\omega}{\kappa}P,\]
so $\nabla\cdot v\in H^1(\Gamma)$ as long as $\kappa$ is at least $C^1$.

It would be more satisfying (and useful in other contexts) to have a regularity theory derived from the weak form of the equations presented herein, and this is a current topic of inquiry for the authors.

\section{Euler-Lagrange Equation for the Model Problem}\label{el_section}
For our model in the development of the numerical method, we will focus on the Dirichlet problem with functional
\[\hat Y(u',G')=\int_\Gamma \left(\begin{array}{c}
\sqcap u' \\
-\mathcal{G}' \\
\end{array}\right)\cdot\mathcal{L}\left(\begin{array}{c}
\sqcap u' \\
-\mathcal{G}' \\
\end{array}\right)\ dx+2\int_{\partial\Gamma}G'\cdot n u''\ dS.\]
Suppose that $u'$ and $u''$ satisfy (\ref{three}) and $(u',G')$ minimizes $\hat Y$ over all $u'\in u_0'+H_0^1(\Gamma)$ and $G'\in H(\mbox{div},\Gamma)$.  Then if we take any functions $s\in H_0^1(\Gamma)$ and $T\in H(\mbox{div},\Gamma)$ and let $\mathcal{T}=(T,\nabla\cdot T)^T$, we have that
\[\hat Y(u'+ts,G'+tT)=\int_\Gamma\left(\begin{array}{c}
\sqcap u'+t\sqcap s \\
-\mathcal{G}'-t\mathcal{T} \\
\end{array}\right)\cdot\mathcal{L}\left(\begin{array}{c}
\sqcap u'+t\sqcap s \\
-\mathcal{G}'-t\mathcal{T} \\
\end{array}\right) dx\]
\[+2\int_{\partial\Gamma}(G'+tT)\cdot n u''\ dS \]
has a minimum at $t=0$.  Therefore,
\begin{equation}\label{euler-lagrange}0=2\int_\Gamma\left(\begin{array}{c}
\sqcap u' \\
-\mathcal{G}' \\
\end{array}\right)\cdot\mathcal{L}\left(\begin{array}{c}
\sqcap s \\
-\mathcal{T} \\
\end{array}\right)\ dx+2\int_{\partial\Gamma}T\cdot n u''\ dS.\end{equation}
This is the weak form of the equation that we want to solve for $u'$.  In the case of the acoustic equation (\ref{model}), we have
\[u=P,\ L=-\rho^{-1},\ K=0,\  M=\omega^2/\kappa,\ h=0,\ v=(-i/\omega)\rho^{-1}\nabla P,\ G=-i\omega v,\]
so we can rewrite
\begin{align}\label{yhat}\hat Y(P',v'')&=\int_\Gamma\left[\left(\begin{array}{c}
\nabla P' \\
-\omega v'' \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
\nabla P' \\
-\omega v'' \\
\end{array}\right)+\left(\begin{array}{c}
\omega P' \\
-\nabla\cdot v'' \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
\omega P' \\
-\nabla\cdot v''\\
\end{array}\right)\right] dx\notag \\
&+2\int_{\partial\Gamma}\omega v''\cdot nP''\ dS,\end{align}
where $r=-\rho^{-1}$, $k=\kappa^{-1}$, and
\[\mathcal{R}=\left(\begin{array}{cc}
r''+r'(r'')^{-1}r' & r'(r'')^{-1} \\
(r'')^{-1}r' & (r'')^{-1} \\
\end{array}\right),\ \mathcal{K}=\left(\begin{array}{cc}
k''+(k')^2/k'' & k'/k'' \\
k'/k'' & 1/k'' \\
\end{array}\right).\]
The requirement that $Z''$ be positive definite translates to the requirement that
\begin{equation}\label{coerc_bounds}\rho''>\alpha I,\ \kappa''<-\beta,\ \alpha,\beta > 0.\end{equation}

Making the substitutions in (\ref{euler-lagrange}) for the acoustic equation, we find that the Euler-Lagrange equation becomes
\begin{equation}\label{four}0=\int_\Gamma\left[\left(\begin{array}{c}
\nabla P' \\
-\omega v'' \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
\nabla s \\
-\omega T \\
\end{array}\right)+\left(\begin{array}{c}
\omega P' \\
-\nabla\cdot v'' \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
\omega s \\
-\nabla\cdot T \\
\end{array}\right)\right] dx+\int_{\partial\Gamma}\omega T\cdot nP''\ dS\end{equation}
for any $s\in H_0^1(\Gamma)$ and any $T\in H(\mbox{div},\Gamma)$.

\section{Implementation of the Finite Element Method}\label{implement}

Our goal is to test the efficacy of this new variational principle, using a simple, explicit finite element implementation.  Let us assume that $d=2$ and $\Gamma=(0,1)^2$.  In order to find a numerical solution for $P'$, we introduce an $N\times N$ computational grid  with equally spaced nodes $(x_j,y_t)$ for $t,j=1,2,\ldots,N$ and grid spacing $h=1/(N-1)$.  We also introduce the finite element spaces
\[\begin{array}{l}
\displaystyle \Psi=\mbox{span}\left\{\left(1-\frac{|x-x_j|}{h}\right)\left(1-\frac{|y-y_t|}{h}\right)\chi_{tj}:2\le t,j\le N-1\right\} \\
 \Phi_1=\mbox{span}\left\{\left(\begin{array}{c}
\displaystyle \left(1-\frac{|x-x_j|}{h}\right)\left(1-\frac{|y-y_t|}{h}\right) \\
0 \\
\end{array}\right)\chi_{tj}:1\le t,j\le N\right\} \\
\Phi_2=\mbox{span}\left\{\left(\begin{array}{c}
0 \\
\displaystyle \left(1-\frac{|x-x_j|}{h}\right)\left(1-\frac{|y-y_t|}{h}\right) \\
\end{array}\right)\chi_{tj}:1\le t,j\le N\right\} \\
\end{array},\]
where
\[\chi_{tj}(x,y)=\left\{\begin{array}{ll}
1 & \mbox{if } |x-x_j|,|y-y_t|\le h \\
0 & \mbox{otherwise} \\
\end{array}\right. .\]
The bases of each of the finite element spaces are built from simple piecewise bilinear elements.

We can re-index these elements with a single index by setting
\[\begin{array}{l}
\displaystyle \psi_k=\left(1-\frac{|x-x_j|}{h}\right)\left(1-\frac{|y-y_t|}{h}\right)\chi_{tj},\mbox{ where }k=(t-2)(N-2)+j-1,\ k=1,\ldots,(N-2)^2, \\
\phi_{1k}=\left(\begin{array}{c}
\displaystyle \left(1-\frac{|x-x_j|}{h}\right)\left(1-\frac{|y-y_t|}{h}\right) \\
0 \\
\end{array}\right)\chi_{tj}\mbox{ where }k=(t-1)N+j,\ k=1,\ldots,N^2, \\
\phi_{2k}=\left(\begin{array}{c}
0 \\
\displaystyle \left(1-\frac{|x-x_j|}{h}\right)\left(1-\frac{|y-y_t|}{h}\right) \\
\end{array}\right)\chi_{tj}\mbox{ where }k=(t-1)N+j,\ k=1,\ldots,N^2. \\
\end{array}\]
We assume that our finite element solution has the form
\[\left(\begin{array}{c}
P' \\
v'' \\
\end{array}\right)=\left(\begin{array}{c}
\psi_R+\sum_{k=1}^{(N-2)^2}\delta_k\psi_k \\
\sum_{k=1}^{N(N-1)}\beta_k\phi_{1k}+\sum_{k=1}^{N(N-1)}\gamma_k\phi_{2k} \\
\end{array}\right).\]
Here $\psi_R$ is any function that satisfies the desired Dirichlet boundary condition for $P'$.  Making this substitution into (\ref{four}), we get
\begin{flalign}\int_\Gamma & \left[\left(\begin{array}{c}
\sum\delta_k\nabla\psi_k \\
-\omega\sum\beta_k\phi_{1k}-\omega\sum\gamma_k\phi_{2k} \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
\nabla s \\
-\omega T \\
\end{array}\right)\right. & \notag\end{flalign}
\[\left.+\left(\begin{array}{c}
\omega\sum\delta_k\psi_k \\
-\sum\beta_k\nabla\cdot\phi_{1k}-\sum\gamma_k\nabla\cdot\phi_{2k} \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
\omega s \\
-\nabla\cdot T \\
\end{array}\right)\right] dx\]
\begin{flalign} &=-\int_\Gamma\left[\left(\begin{array}{c}
\nabla\psi_0 \\
0 \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
\nabla s \\
-\omega T \\
\end{array}\right)+\left(\begin{array}{c}
\omega\psi_0 \\
0 \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
\omega s \\
-\nabla\cdot T \\
\end{array}\right)\right] dx & -\int_\Gamma\left[\omega\nabla\psi_I\cdot T+\omega\psi_I\nabla\cdot T\right] dx,\notag\end{flalign}
where we have used the divergence theorem on the boundary integral, $\psi_I$ is any function on $\Gamma$ satisfying the desired Dirichlet boundary condition for $P''$, and $s\in H_0^1(\Gamma),\ T\in H(\mbox{div},\Gamma)$ are arbitrary.  In particular, this must hold when
\[s=\psi_k,\ T=0\mbox{ for }k=1,\ldots,(N-2)^2\]
\[s=0,\ T=\phi_{1k}\mbox{ for }k=1,\ldots,N(N-1)\]
\[s=0,\ T=\phi_{2k}\mbox{ for }k=1,\ldots,N(N-1).\]
This gives rise to a system of equations of the form $A\alpha=b$, where $A$ has the block form
\begin{equation}\label{block}A=\left(\begin{array}{ccc}
A_1 & A_4 & A_6 \\
A_4 & A_2 & A_5 \\
A_6 & A_5 & A_3 \\
\end{array}\right)\end{equation}
and the blocks have entries
\begin{eqnarray}\label{block_entries}(A_1)_{tj}&=&\displaystyle\int_\Gamma\left[\left(\begin{array}{c}
\nabla \psi_t \\
0 \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
\nabla\psi_j \\
0 \\
\end{array}\right)+\left(\begin{array}{c}
\omega\psi_t \\
0 \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
\omega\psi_j \\
0 \\
\end{array}\right)\right] dx \notag \\
(A_2)_{tj}&=&\displaystyle\int_\Gamma\left[\left(\begin{array}{c}
0 \\
-\omega\phi_{1t} \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
0 \\
-\omega\phi_{ij} \\
\end{array}\right)+\left(\begin{array}{c}
0 \\
-\nabla\cdot\phi_{1t} \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
0 \\
-\nabla\cdot\phi_{1j} \\
\end{array}\right)\right] dx \notag \\
(A_3)_{tj}&=&\displaystyle\int_\Gamma\left[\left(\begin{array}{c}
0 \\
-\omega\phi_{2t} \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
0 \\
-\omega\phi_{2j} \\
\end{array}\right)+\left(\begin{array}{c}
0 \\
-\nabla\cdot\phi_{2t} \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
0 \\
-\nabla\cdot\phi_{2j} \\
\end{array}\right)\right] dx  \\
(A_4)_{tj}&=&\displaystyle\int_\Gamma\left[\left(\begin{array}{c}
0 \\
-\omega\phi_{1t} \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
\nabla\psi_j \\
0 \\
\end{array}\right)+\left(\begin{array}{c}
0 \\
-\nabla\cdot\phi_{1t} \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
\omega\psi_j \\
0 \\
\end{array}\right)\right] dx. \notag \\
(A_5)_{tj}&=&\displaystyle\int_\Gamma\left[\left(\begin{array}{c}
0 \\
-\omega\phi_{2t} \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
0 \\
-\omega\phi_{1j} \\
\end{array}\right)+\left(\begin{array}{c}
0 \\
-\nabla\cdot\phi_{2t} \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
0 \\
-\nabla\cdot\phi_{1j} \\
\end{array}\right)\right] dx \notag \\
(A_6)_{tj}&=&\displaystyle\int_\Gamma\left[\left(\begin{array}{c}
0 \\
-\omega\phi_{2t} \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
\nabla\psi_j \\
0 \\
\end{array}\right)+\left(\begin{array}{c}
0 \\
-\nabla\cdot\phi_{2t} \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
\omega\psi_j \\
0 \\
\end{array}\right)\right] dx. \notag
\end{eqnarray}
The right-hand side vector $b$ is partitioned as
\[b=\left(\begin{array}{c}
b_1 \\
b_2 \\
b_3 \\
\end{array}\right),\]
where
\begin{equation}\label{five}\begin{array}{ll}
(b_1)_k= & -\displaystyle\int_\Gamma\left[\left(\begin{array}{c}
\nabla\psi_R \\
0 \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
\nabla\psi_k \\
0 \\
\end{array}\right)+\left(\begin{array}{c}
\omega\psi_R \\
0 \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
\omega\psi_k \\
0 \\
\end{array}\right)\right] dx \\
(b_2)_k= & -\displaystyle\int_\Gamma\left[\left(\begin{array}{c}
\nabla\psi_R \\
0 \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
0 \\
-\omega\phi_{1k} \\
\end{array}\right)+\left(\begin{array}{c}
\omega\psi_R \\
0 \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
0 \\
-\nabla\cdot\phi_{1k} \\
\end{array}\right)\right] dx \\
 & -\displaystyle\int_\Gamma\left[\omega\nabla\psi_I\cdot\phi_{1k}+\omega\psi_I\nabla\cdot\phi_{1k}\right] dx \\
(b_3)_k= & -\displaystyle\int_\Gamma\left[\left(\begin{array}{c}
\nabla\psi_R \\
0 \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
0 \\
-\omega\phi_{2k} \\
\end{array}\right)+\left(\begin{array}{c}
\omega\psi_R \\
0 \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
0 \\
-\nabla\cdot\phi_{2k} \\
\end{array}\right)\right] dx \\
 & -\displaystyle\int_\Gamma\left[\omega\nabla\psi_I\cdot\phi_{2k}+\omega\psi_I\nabla\cdot\phi_{2k}\right] dx. \\
\end{array}\end{equation}

The method for solving for $P'$ and $v''$ can be easily modified to solve for $P''$ and $v'$.  In this case the weak equation is
\[\int_\Gamma\left[\left(\begin{array}{c}
\nabla P'' \\
\omega v' \\
\end{array}\right)\cdot\mathcal{R}\left(\begin{array}{c}
\nabla s \\
\omega T \\
\end{array}\right)+\left(\begin{array}{c}
-\omega P'' \\
-\nabla\cdot v' \\
\end{array}\right)\cdot\mathcal{K}\left(\begin{array}{c}
-\omega s \\
-\nabla\cdot T \\
\end{array}\right)\right] dx+\int_{\partial\Gamma}\omega T\cdot nP'\ dS,\]
and all the methods above still apply.  In fact, to obtain the new matrix for this formulation, we simply change the signs of the blocks $A_4$ and $A_6$, and the changes in $b$ are mostly reversing signs and the roles of the two auxiliary functions $\psi_R$ and $\psi_I$.

\subsection{Other Discretizations}\label{other_disc}
Along with the discretization described in Section~\ref{implement}, we have experimented with two other implementations in which different basis functions are used to represent the variable $v$.  The first of these uses the Raviart-Thomas $RT_{[0]}$ elements described in \cite{Brezzi-Fortin}.  We found that the resulting finite element matrix is much more poorly scaled, with a condition number approximately twice as large as that obtained with the nodal bilinear basis.  The second method uses the $RT_{[1]}$ elements (also described in \cite{Brezzi-Fortin}).  In this case, the higher-order basis functions obviously result in a somewhat less-sparse finite element matrix, and the condition number is approximately the same as that obtained with the all-bilinear discretization.

\section{Conditioning}\label{conditioning_section}

 \begin{figure}[tbp]
   \centering
   \includegraphics[width=7in]{eigenvalues.jpg}
   \caption{The distribution of the eigenvalues of $A$ and the real parts of the eigenvalues of $M^{-1}A$ for $N=30$.}
   \label{evals}
\end{figure}

As was mentioned, perhaps the greatest numerical advantage to having a minimization formulation for the Helmholtz equation is that the matrix produced by the finite element method is symmetric positive definite.  This allows for the use of methods such as the conjugate gradient method to solve the system.  Of course, the use of a preconditioning matrix in the conjugate gradient method can speed up the convergence considerably, which is especially important when solving the relatively large sparse systems generated by the finite element approach outlined above.

In our approach, there are three basic types of elements used: bilinear elements, first component bilinear vector elements, and second component bilinear vector elements.  Each of these types of elements interacts with all of the other types, and these interactions are what give rise to the blocks in (\ref{block}).  Assuming that interactions among similar element types are most important, we choose the block Jacobi preconditioner
\[M=\left(\begin{array}{ccc}
A_1 & 0 & 0 \\
0 & A_2 & 0 \\
0 & 0 & A_3 \\
\end{array}\right).\]
Among all block diagonal preconditioners of this form, this choice of $M$ minimizes the condition number of $M^{-\frac{1}{2}}AM^{-\frac{1}{2}}$ to within a factor of 3 of its minimum \cite{Demmel}.

As one of the steps in the preconditioned conjugate gradient method (PCG), \cite{Demmel_book}, a system of the form $Mr=y$ must be solved.  In order to make solving this problem more efficient, we precondition the matrix $M$ and use conjugate gradient to solve this system as well.  The preconditioner used in this inner implementation of PCG was an incomplete Choleski factorization of M.  Figure \ref{evals} shows the distribution of the eigenvalues of the matrix $A$ before and after preconditioning for $N=30$.    In Figure \ref{iters}, we see the how the number of PCG iterations grows with $N$ for several error tolerances.

 \begin{figure}[tbp]
   \centering
   \includegraphics[width=4in]{bilin_pcg_iters}
   \caption{The growth of the number of PCG iterations required to solve a given problem with grid size for several error tolerances (outer implementation of PCG only).}
   \label{iters}
\end{figure}

A  key component in ensuring that the system $A\alpha=b$ is well conditioned is for the matrix $\mathcal{L}$ (or equivalently $\mathcal{R}$ and $\mathcal{K}$) to have a coercivity constant that is as large as possible.  For this reason, we expect better numerical results when the eigenvalues of $\mathcal{L}$ are bounded well away from zero. In the case of the Helmholtz equation, the matrix $Z$ is diagonal, say $Z=\mbox{diag}(c_1,\ldots,c_{d+1})$, which makes it possible to calculate the eigenvalues of $\mathcal{L}$.  If $D$ is an invertible matrix, then we may factor a block diagonal matrix
\[\left(\begin{array}{cc}
A & B \\
C & D \\
\end{array}\right)\]
as
\[\left(\begin{array}{cc}
A & B \\
C & D \\
\end{array}\right)=\left(\begin{array}{cc}
I & B \\
0 & D \\
\end{array}\right)\left(\begin{array}{cc}
A-BD^{-1}C & 0 \\
D^{-1}C & I \\
\end{array}\right),\]
which implies that
\[\mbox{det}\left(\begin{array}{cc}
A & B \\
C & D \\
\end{array}\right)=\mbox{det}(D)\mbox{det}(A-BD^{-1}C).\]
Therefore,
\[\mbox{det}(\mathcal{L}-\lambda I)=(-1)^{d+1}\mbox{det}\left(\begin{array}{cc}
(Z'')^{-1}Z' & (Z'')^{-1}-\lambda I \\
Z''+Z'(Z'')^{-1}Z'-\lambda I & Z'(Z'')^{-1} \\
\end{array}\right)\]
\[=(-1)^{d+1}\mbox{det}(Z'(Z'')^{-1})\mbox{det}((Z'')^{-1}Z'+[-(Z')^{-1}+\lambda Z''(Z')^{-1}][Z''+Z'(Z'')^{-1}Z'-\lambda I])\]
\[=(-1)^{d+1}\mbox{det}(Z'(Z'')^{-1})\mbox{det}(\lambda^2[-Z''(Z')^{-1}]+\lambda[(Z')^{-1}+Z''(Z')^{-1}Z''+Z']-(Z')^{-1}Z'').\]
In the case of diagonal $Z$, this implies that
\[\lambda=\frac{-a_j\pm\sqrt{a_j^2-b_j^2}}{-b_j}\ \ \ j=1,\ldots,d+1,\]
where
\[a_j=\frac{1}{c_j'}+\frac{(c_j'')^2}{c_j'}+c_j'\ \mbox{ and }\ b_j=2\frac{c_j''}{c_j'}.\]
If $Z'=0$, then $\mathcal{L}$ is diagonal, and its eigenvalues are those of $Z''$ and $(Z'')^{-1}$.

The above analysis tells us that the finite element problem will be better conditioned for those problems where the coefficients $\rho$ and $\kappa$ are such that $Z$ is close to $Ii$, i.e. $\rho=iI$ and $\kappa=-Ii$ (this would correspond to the limiting case where $a_j=b_j$).  In many cases when we are presented with a problem where the coercivity constant for $\mathcal{L}$ is small, we can apply an appropriate rotation and scaling to the problem in order to get a finite element matrix that is better conditioned.  By multiplying the problem (\ref{model}) through by a complex constant $re^{i\theta}$, we effectively replace $Z$ with $re^{i\theta}Z$, so we should choose $r$ and $\theta$ so that $re^{i\theta}Z$ is as close as possible to $iI$.  However, this may not always be possible, for example, when an isotropic $\rho(x)$ oscillates between values in the upper half of the complex plane that are close to $1$ and $-1$.

\section{Numerical Results}\label{numerical_section}

As an example, we demonstrate the error bound on the problem (\ref{acoustic}), with parameters $\rho=(-5+5i)I,\ \kappa=4-4i$ and $\omega=2$.  A solution is $P(x,y)=e^{2ix-3y}$.  In this example we took
\[\psi_R=\mbox{Re}(e^{2ix-3y})+\sin(\pi x)\sin(\pi y),\ \psi_I=\mbox{Im}(e^{2ix-3y})+\sin(\pi x)\cdot 3\sin(\pi y))\]
and solved the problem on grids with $N=3,\ldots,100$.  Table 1 shows the error in the finite element solution for various values of $N$.  The errors were calculated using the trapezoidal rule with function evaluations on a grid with size $N=1500$.  Figure~\ref{example_fig} demonstrates the method on a problem with non-constant coefficients, where the dissipation in the material is higher inside a disk centered in the unit square.  The boundary conditions for the real part are oscillatory, while the boundary conditions for the imaginary part are simply an affine function.

\begin{table}
\label{error_table}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
N & h & $\|(P-P_N,v-v_N)\|_V$ \\
\hline
30 & 0.0345 & $6.6162\times 10^{-4} $\\
40 & 0.0256 & $3.6692\times 10^{-4}$ \\
50 & 0.0204 & $ 2.3252\times 10^{-4}$ \\
60 & 0.0169 & $1.6026\times 10^{-4}$ \\
70 & 0.0145 & $1.1722\times 10^{-4}$ \\
80 & 0.0127 & $8.9706\times 10^{-5}$ \\
90 & 0.0112 & $7.0686\times 10^{-5} $\\
100 & 0.0101 & $5.7037\times 10^{-5}$ \\
\hline
\end{tabular}
\end{center}
\caption{The error in the finite element solution for various values of the grid size $N$.}
\end{table}

 \begin{figure}[t]
   \centering
   \includegraphics[width=2.953in]{P_.jpg}
   \includegraphics[width=2.953in]{P__.jpg}
   \caption{The solutions $P'$ and $P''$,  with $\omega=10,\psi_R=\sin(6\pi x)\cos(3\pi y),\ \psi_I=3x+5y+2,\ N=30$.  There is a circular inclusion in the center of the domain with  $\rho=.01+.001i$ and $\kappa=.01-.003i$ outside the inclusion and $\rho=-5+5i$ and $\kappa=4-4i$ inside the inclusion.}
     \label{example_fig}
\end{figure}

\section{Robin Boundary Conditions}\label{robin}

\subsection{Problem Formulation}

Another boundary condition that often appears is the Robin problem
\[\left\{\begin{array}{ll}
\displaystyle -\nabla\cdot\rho^{-1}\nabla P-\frac{\omega^2}{\kappa}P=0 & \mbox{in}\ \Gamma, \\
P+av\cdot n=g & \mbox{on}\ \partial\Gamma, \\
\end{array}\right.\]
where $a\in\mathbb{C}$.  In order to deal with this boundary condition, which concerns both real and imaginary parts of the variables $P$ and $v$ simultaneously, we start with the minimization functional for the natural boundary conditions
\[Y(P',v'')+2\omega\int_{\partial\Gamma}\left[P'v'\cdot n+P''v''\cdot n\right]dS.\]
The Euler-Lagrange Equation for the corresponding variational principle is
\[B(P',v'',s,T)=-\omega\int_{\partial\Gamma}\left[sv'\cdot n+P''T\cdot n\right] dS.\]
where the bilinear form $B$ is defined in (\ref{bilinear}).  Notice that we can write the surface integral above as
\[-\omega\int_{\partial\Gamma}\left(\begin{array}{c}
v'\cdot n \\
P'' \\
\end{array}\right)\cdot\left(\begin{array}{c}
s \\
T\cdot n \\
\end{array}\right)\ dS.\]
The vector on the right contains the primary variables for which we would like to solve, and the vector on the left contains the dual variables which we would like to eliminate using the Robin boundary condition.  In terms of the vectors above, we can express the Robin condition as
\[M_1\left(\begin{array}{c}
P' \\
v''\cdot n \\
\end{array}\right)+M_2\left(\begin{array}{c}
v'\cdot n \\
P'' \\
\end{array}\right)=\left(\begin{array}{c}
g' \\
g'' \\
\end{array}\right),\]
where
\[M_1=\left(\begin{array}{cc}
1 & -a'' \\
0 & a' \\
\end{array}\right)\ \mbox{and}\ M_2=\left(\begin{array}{cc}
a' & 0 \\
a'' & 1 \\
\end{array}\right).\]
Rearranging, we find that
\[\left(\begin{array}{c}
v'\cdot n \\
P'' \\
\end{array}\right)=M_2^{-1}\left(\begin{array}{c}
g' \\
g'' \\
\end{array}\right)-M_2^{-1}M_1\left(\begin{array}{c}
P' \\
v''\cdot n \\
\end{array}\right),\]
so the surface integral term becomes
\[-\omega\int_{\partial\Gamma}\left[M_2^{-1}\left(\begin{array}{c}
g' \\
g'' \\
\end{array}\right)-M_2^{-1}M_1\left(\begin{array}{c}
P' \\
v''\cdot n \\
\end{array}\right)\right]\cdot\left(\begin{array}{c}
S \\
T\cdot n \\
\end{array}\right)\ dS\]
\[=-\omega\int_{\partial\Gamma}M_2^{-1}\left(\begin{array}{c}
g' \\
g'' \\
\end{array}\right)\cdot\left(\begin{array}{c}
P' \\
v''\cdot n \\
\end{array}\right)\ dS+\omega\int_{\partial\Gamma}M_2^{-1}M_1\left(\begin{array}{c}
P' \\
v''\cdot n \\
\end{array}\right)\cdot\left(\begin{array}{c}
P' \\
v''\cdot n \\
\end{array}\right)\ dS.\]
The new Euler-Lagrange equation for the Robin boundary condition is therefore
\[B(P',v'';s,T)-\omega\int_{\partial\Gamma}M_2^{-1}M_1\left(\begin{array}{c}
P' \\
v''\cdot n \\
\end{array}\right)\cdot\left(\begin{array}{c}
s \\
T\cdot n \\
\end{array}\right)\ dS\]
\[=-\omega\int_{\partial\Gamma}M_2^{-1}\left(\begin{array}{c}
g' \\
g'' \\
\end{array}\right)\cdot\left(\begin{array}{c}
s \\
T\cdot n \\
\end{array}\right)\ dS.\]
Since
\[M_2^{-1}=\frac{1}{a'}\left(\begin{array}{cc}
1 & 0 \\
-a'' & a' \\
\end{array}\right),\]
we have
\[M_2^{-1}M_1=\frac{1}{a'}\left(\begin{array}{cc}
1 & -a'' \\
-a'' & |a|^2 \\
\end{array}\right),\]
which is positive definite as long as $a'>0$.  The new bilinear form above is guaranteed to be coercive as long as $\rho$ and $\kappa$ satisfy (\ref{coerc_bounds}) and $a'<0$.

To find a numerical solution for the Robin boundary value problem, we discretize using the finite element scheme presented in Section~\ref{implement}.  Unfortunately, the surface integrals can no longer be converted to volume integrals by integration by parts and must be computed as they stand.  In this case, the finite element matrix is written as the sum of two matrices $A-\omega B$, where $A$ is of the form (\ref{block}), and the blocks have entries (\ref{block_entries}), and another matrix $B$ with the same block form and block entries
\begin{eqnarray*}(B_1)_{tj}&=&\displaystyle\int_{\partial\Gamma}\left(\begin{array}{c}
 \psi_t \\
0 \\
\end{array}\right)\cdot M_2^{-1}M_1\left(\begin{array}{c}
\psi_j \\
0 \\
\end{array}\right)\ dS \\
(B_2)_{tj}&=&\displaystyle\int_{\partial\Gamma}\left(\begin{array}{c}
0 \\
\phi_{1t}\cdot n \\
\end{array}\right)\cdot M_2^{-1}M_1\left(\begin{array}{c}
0 \\
\phi_{1j}\cdot n \\
\end{array}\right)\ dS \\
(B_3)_{tj}&=&\displaystyle\int_{\partial\Gamma}\left(\begin{array}{c}
0 \\
\phi_{2t}\cdot n \\
\end{array}\right)\cdot M_2^{-1}M_1\left(\begin{array}{c}
0 \\
\phi_{2j}\cdot n \\
\end{array}\right)\ dS \\
(B_4)_{tj}&=&\displaystyle\int_{\partial\Gamma}\left(\begin{array}{c}
0 \\
\phi_{1t}\cdot n \\
\end{array}\right)\cdot M_2^{-1}M_1\left(\begin{array}{c}
\psi_j \\
0 \\
\end{array}\right)\ dS \\
(B_5)_{tj}&=&\displaystyle\int_{\partial\Gamma}\left(\begin{array}{c}
0 \\
\phi_{2t}\cdot n \\
\end{array}\right)\cdot M_2^{-1}M_1\left(\begin{array}{c}
0 \\
\phi_{1j}\cdot n \\
\end{array}\right)\ dS \\
(B_6)_{tj}&=&\displaystyle\int_{\partial\Gamma}\left(\begin{array}{c}
0 \\
\phi_{2t}\cdot n \\
\end{array}\right)\cdot M_2^{-1}M_1\left(\begin{array}{c}
\psi_j \\
0 \\
\end{array}\right)\ dS. \\
\end{eqnarray*}
The right-hand side vector $b$ is also partitioned as $(b_1,b_2,b_3)^T$ with entries
\[\begin{array}{ll}
(b_1)_k= & \displaystyle-\omega\int_{\partial\Gamma}\left(\begin{array}{c}
\psi_k \\
0 \\
\end{array}\right)\cdot M_2^{-1}\left(\begin{array}{c}
g' \\
g'' \\
\end{array}\right)\ dS \\
(b_2)_k= & -\displaystyle\omega\int_{\partial\Gamma}\left(\begin{array}{c}
0 \\
\phi_{1k}\cdot n \\
\end{array}\right)\cdot M_2^{-1}\left(\begin{array}{c}
g' \\
g'' \\
\end{array}\right)\ dS \\
 (b_3)_k= & -\displaystyle\omega\int_{\partial\Gamma}\left(\begin{array}{c}
0 \\
\phi_{2k}\cdot n \\
\end{array}\right)\cdot M_2^{-1}\left(\begin{array}{c}
g' \\
g'' \\
\end{array}\right)\ dS. \\
\end{array}\]

Assuming that the coercivity requirements (\ref{coerc_bounds}) on $\rho$ and $\kappa$ are satisfied, and $a' < 0$, the system
\[(A-\omega B)\alpha=b\]
may be solved using the same preconditioned conjugate gradient approach as outlined previously.

 \begin{figure}[t]
   \centering
   \includegraphics[width=2.8994in]{circle1.jpg}
   \includegraphics[width=2.8994in]{bar1.jpg}
   \caption{On the left is shown the solution to the Robin problem with a disc of high density material centered in the domain.  On the right the disk is replaced with a bar of the same material angled from the lower left to the upper right of the domain.}
     \label{circ_scat}
\end{figure}

\subsection{Numerical Examples}

Here we present some numerical examples obtained by using the finite element method to solve problems with Robin boundary conditions.  In these examples the Robin boundary conditions are imposed on $y=0$ and $y=1$, while on the other sides of the domain we have imposed periodic boundary conditions.  On the left in Figure~\ref{circ_scat} is the  solution with a circular scatterer with $\rho=1+.011i$ outside the scatterer, $\rho=2+.011i$ inside the scatterer, $\kappa=1+.011i$ everywhere, $a=-1+.333i$ and $g=3.33i$.  On the right, the circular scatterer is replaced by a bar angled across the domain, but the other parameters in the problem remain the same.  These results were calculated using the $RT_{[0]}$ discretization for the $v$ variable described in Section~\ref{other_disc}.

\section{Conclusions}
The variational principles of Milton, Seppecher, Bouchitt\'{e}, and Willis make it possible to formulate the solution of the Helmholtz equation as a minimization, and this is reflected in the fact that the stiffness matrix for the finite element method is symmetric positive definite.  This allows us to use classical iterative methods such as preconditioned conjugate gradient to solve the associated system, along with straightforward finite element error estimates.  The primary advantage of this approach is that it allows the use of efficient iterative methods for the solution of the linear system. But there are also disadvantages in that the system has more unknowns, since we must solve for $P$ and $v$ simultaneously.

More research is necessary to determine the circumstances under which this approach may be more effective than others currently in use.  A particular point of interest is that even though the underlying minimization principles are valid for arbitrarily small loss coefficients, the conditioning of the associated finite element matrix deteriorates as the system becomes less dissipative.  The general question of how solution efficiency depends on loss should be analyzed further.

We have only approached the scalar, two-dimensional Helmholtz equation in this paper, while the minimization principles of Milton, Seppecher, Bouchitt\'{e}, and Willis apply to the full vector Maxwell equations, as well as the equations of linear elasticity.  The general approach taken here should also apply in those cases.  We note finally that in many applications for these models, one would like to apply nonlocal transmission or radiation boundary conditions in order to accurately handle unbounded domains.  The problem of adapting these minimization methods to such boundary conditions remains open, although presumably the PML approach (see eg. \cite{PML}) would apply.

\section*{Acknowledgements}

Russell Richins is grateful for support from the National Science Foundation through grant DMS-0707978.  Also, the authors would like to thank Graeme Milton for many helpful suggestions, and John Willis who, along with Graeme Milton, clarified an earlier formulation of the boundary value problem for us.

\bibliographystyle{amsplain}


text:
Radnice Prahy 7 je budova, kde sídlí úřad městské části Prahy 7 a nachází se v Holešovicích. Byla otevřena v roce 2020, vznikla jako přestavba objektu, který stál na jejím místě původně. V této stavbě je celkem 202 pracovních míst a zasedací místnosti pro 230 lidí.

Historie

Plány
Praha 7 jako jedna z mála pražských městských částí neměla do postavení nového objektu v roce 2020 vlastní radnici. Od roku 1993 provizorně sídlila v budově Generálního finančního ředitelství na Nábřeží kapitána Jaroše. Před rokem 1993 působila radnice v provizorní budově zvané "dřevák" u Parkhotelu na Letné. Původní nájemní smlouvu platnou do konce roku 2015 krátce před vypršením radnice prodloužila do roku 2019.

Městská část Praha 7 na začátku plánovala výstavbu zcela nové budovy radnice. Vedení Prahy 7 skládající se z ODS chtělo postavit budovu za 1,6 miliardy korun – na místě, kde byla později postavena budova Art Gen. Pedagog Jan Čižinský v roce 2013 inicioval referendum, aby byly náklady na novou budovu omezeny na 500 milionů korun. O rok později po volbách se Jan Čižinský se svým sdružením Praha 7 sobě stal starostou sedmé městské části.

Odkoupení a výstavba
Praha 7 budovu radnice v ulici U Průhonu 38 z 90. let 20. století koupila v roce 2015 za 90 milionů na základě veřejné soutěže. V roce 2016 byla vyhlášena na návrh přestavby tohoto objektu veřejná architektonická soutěž. Přihlásilo se do ní 70 různých soutěžících. Vyhrál český Atelier bod architekti, konkrétně architekti Vojtěch Sosna, Jáchym Svoboda a Jakub Straka. Stavbu provádělo konsorcium Metrostav-Geosan Group.

Dokončení rekonstrukce bylo plánováno na rok 2019, kvůli náročnosti přestavby budovy byla nakonec dokončena v lednu 2020, kdy proběhla kolaudace. K oficiálnímu otevření došlo na začátku 2020, úřady zde budou fungovat od března téhož roku. Za výstavbu a vybavení budovy městská část zaplatila přibližně 240 milionů korun bez DPH.

Popis
Vzhled přestavěného objektu cituje a odráží "tvarosloví" okolní čtvrti, Holešovic. Fasáda je z červených cihel. Z původního objektu na stavbě zůstal pouze skelet, tedy nosná konstrukce. Interiéry jsou navrženy minimalisticky a transparentně. Každé patro má pro lepší orientaci jinak barevnou podlahu. Dveře kanceláří mají vždy skleněné okno. Kromě kanceláří se zde zastupitelský sál či konferenční místnosti. Na střeše je přístupná terasa a radniční sál pro konání kulturních akcí. Na nejvyšším bodu fasády se nachází také hodiny, které dodala firma Ludvig Heinz, sídlící nedaleko v Holešovicích.

V listopadu roku 2021 získal Atelier bod architekti s budovou radnice prestižní cenu Grand Prix Architektů - Národní cena za architekturu v kategorii renovace.

Odkazy

Reference

Externí odkazy

 Atelier bod architekti - stránka projektu radnice Prahy 7

Moderní architektura v Praze
Postaveno v Česku 2020
Stavby v Holešovicích (Praha 7)
7

redpajama_set_name: RedPajamaWikipedia


Closed lanes, tailbacks, delays, injury and even loss of life are just some of the tyre related consequences that have inspired a new set of advisory messages to drivers travelling on the UK's motorways.
Women drivers are taking unnecessary risks on the road because they lack confidence in simple car maintenance, research from Hankook Tyre UK Ltd has found.
As the heatwave continues ahead of the great summer get-away, drivers are being reminded about the need to make sure their tyres are properly inflated, to avoid temperature induced tyre failures.
A new recycling initiative is giving motorists the chance to help the planet when it comes to changing their worn out tyres.
With the news that Britain faces up to 10 years of dismally wet summers, TyreSafe, the UK's leading tyre safety organisation, is advising motorists that adequate tyre tread depth is more important than ever.
Following last year's UK summer, which proved to be the wettest on record, Micheldever Tyre Services/Protyre is advising motorists before they set off on a summer break to check their tyres to be safe and secure on the road.
Time to re-tyre your rubber?
Over 10 million tyres on UK roads could be dangerous, not because of poor tread, but because of their age, according to new research.
Respected Japanese manufacturer Toyo Tires has launched an exciting new tyre design that promises the highest levels of safety, grip and wet-weather performance in an attractive case design that comes in many sizes and fitments.
Respected Japanese tyre manufacturer Toyo Tires has announced its support of the British Army Motorsport Association (BAMA).
After one of the longest and coldest winters in recent memory, the UK's 650,000 touring caravan and motorhome owners are being reminded about the importance of checking their vehicle's tyres before hitting the roads this spring.
Respected Japanese tyre manufacturer Toyo continues a successful start to the 2013 race season by being chosen as the control tyre for two more leading FWD championships.
Tech savvy youngsters may be able to master the myriad functions of a smartphone or tablet, but when it comes to knowing and maintaining their cars, OAPs leave younger drivers standing at the lights.
More than half of drivers have no idea whether their car has a spare wheel and even more wouldn't feel confident enough to change one if they experienced a puncture according to a survey by Halfords Autocentres.
Delivering superior performance is engrained in the genetic code of the AMG brand. It's also the mission assigned to the new Michelin Pilot Sport Cup 2 tyres, which will equip all new Mercedes-Benz SLS AMG Coupe Black Series cars.


text:
Karbonn, Lava and Xolo set to launch Windows Phone devices in the next few months
KarbonnLavaXOLO
By Jonathan Pereira	 On Mar 17, 2014 Last updated Mar 17, 2014
We had recently seen reports suggesting that some of our well-known Indian manufacturers would soon be entering the Windows Phone segment. Confirming these reports nearly three companies have revealed their plans.
According to sources, as many as three Indian smartphone companies would be starting their journeys in the Windows Phone industry by the end of the first half of 2014. The three companies to lead are Karbonn, which is said to launch Android and Windows dual boot devices, and Lava and Xolo.
Interestingly, most of the Windows Phone devices from these companies would start the price range of Rs. 6000 and may go up to Rs. 12000. This could be convincing factor for users to jump to the Windows Phone bandwagon. However given the pricing, we cannot expect any high end specs but more of entry to mid-range devices.
Confirming their venture, Karbonn Managing Director Pradeep Jain said, "We will come out with four devices with Windows-based operating system by the end of May or starting of June in the price range of 6,000-12,000"
Similarly, Hari Om Rai, CEO, Lava International said, "A series of Windows-based phones will be launched under the Lava brand by July. These will be priced in the range of Rs 6,500-8,500. We expect the Windows-based phones to represent 8-10% volume of Lava smartphones starting July-August time- frame."
Samsung ATIV SE Windows Phone handset with Galaxy S4 design leaked
Xiaomi Red Rice Note with 5.5 inch display and octa core processor teased

redpajama_set_name: RedPajamaCommonCrawl


text:
Elke Karsten (født 15. maj 1995 i Corrientes, Argentina) er en argentinsk håndboldspiller, der spiller for BM Bera Bera, i Spanien.

Referencer

Eksterne henvisninger

Håndboldspillere fra Argentina
Håndboldspillere fra BM Bera Bera
Olympiske håndboldspillere fra Argentina
Håndboldspillere ved sommer-OL 2016
Deltagere for Argentina ved sommer-OL 2016


text:
Kilcoyne AccountantsBlogSyndicatedCo-operative housing delivers record 454 homes in 2020
Co-operative housing delivers record 454 homes in 2020
The national organisation promoting and developing co-operative housing in Ireland said it delivered 454 homes last year, despite the challenges posed by Covid-19.
Co-operative Housing Ireland said that represented an increase of over a fifth on the previous year and that it was a record number of housing deliveries for the organisation.
The outcome was achieved despite a shutdown of the construction sector for several weeks from late March to early May last year at the onset of the Covid-19 pandemic here.
According to its annual report, to be published later today, CHI said the homes were delivered across 13 counties in cities and small towns.
The majority of the homes – 236 – were delivered in Munster with 199 in Leinster and the remaining 19 in Connacht.
"The housing need is nationwide and so too must be our response. To achieve this landmark delivery €111.2 million of loans were drawn down in 2020. We are on target to deliver more than 500 homes this year," Chairperson of CHI Pearse O'Shiel said.
"The pandemic further highlighted that the only guaranteed affordable-rented homes in Ireland are provided by local authorities, housing associations, and housing co-operatives," he added.
The publication of CHI's annual report coincides with the launch today of a development of 80 homes in County Kildare, the largest CHI development in the county to date.
Loughlion Green, which is located close to Kildare Village, is a development of A-rated social-rented homes ranging in size form one to four bedroom units.
The scheme was developed through a partnership with Kildare County Council, Bank of Ireland, the Department of Housing, Local Government and Heritage, and Kelby Developments.
Article Source – Co-operative housing delivers record 454 homes in 2020 – RTE – Brian Finn


Due to recently enacted legislation in Indiana, a new petition has been posted at Change.org proposing a move of Gen Con from Indianapolis to Ohio. Gen Con has been in Indianapolis since 2003 and has signed an extension with Visit Indy through 2020.
With the new law enacted in Indiana this past week, that state has shown it no longer wants to hold events inclusive of all peoples. Gamers come from all walks of life, socio-economic backgrounds, religious faiths and sexual orientations. The state of Indiana has now resoundingly said to these people "We don't want your business." and we should be more than happy to oblige. I suggest Ohio as a good place to relocate too. Cities such as Columbus, Toledo, Cleveland, Cincinnati ect would be more than happy to host such an event. Also a move to Ohio wouldn't amount to much of a long distance from where the convention is currently held.
Please keep the comments civil or we will close comments on this post immediately.
Shapeways – Digital 3D Printing Services for Tabletop Games!


I was so happy to see this post. A lot of the other blogs I read haven't mentioned the grape season yet, it seems a lot of people skipped onto apple season already. I grew up near grape valleys (Concord Grape Pie mmm) and now that I live deeply entrenched in a food desert city, I miss the satisfaction from group effort harvesting in orchards. Beautifully written and visioned, this post brought back lots of nostalgia for me. Thanks.
Anything to cause people to reflect on their glass, some distant evening from a previous vintage, is really important. All of the hands that touch and encourage wine into it's being, from planting, pruning, managing all of the variants that make or break a year's worth of nature-plus-people working it out together to raise the collective awareness of that magical communion is important. What you have written here is important.
There's a wonderful connectedness to the cycle of the growing season, culminating in the immediacy of the moment of harvest. And, what can be more satisfying than creating wine? I love your description of the violet-red creases in Ben's palms– that's a poignant recognition of how much he becomes an integral part of the whole process. And I think it is that which makes this kind of work so truly meaningful. Yes, gardening is perhaps the closest we can get–food gardening, especially if our own food security is involved. I love the entire process, from dreaming and planning to all the work of planting, nurturing, harvesting, enjoying preparing nourishing meals and preserving food for the Winter. I would think that your field work, often involving travel to diverse cultures, has some similar attributes. Sarah, I hope so.
Its just, a little crush, da da da da , every time we touch…..!
Come on on that's some classic tuneidge!
That song never does anyone any wrong!
You are currently reading Crush at The Yellow House.


SANTA MONICA, Calif. (January 15, 2014) — The Recording Academy® Producers & Engineers Wing® announces additional sponsors for its seventh annual GRAMMY® Week event honoring two-time GRAMMY® Award winner Neil Young for his commitment to excellence and ongoing support for the art and craft of recorded music. The event will be held on Tuesday, Jan. 21, 2014, at The Village Studios in West Los Angeles. GRAMMY Week culminates with the 56th Annual GRAMMY Awards® on Sunday, Jan. 26, 2014, airing live on the CBS Television Network, at 8 p.m. ET/PT.
Newly announced sponsors for the event include Silver Sponsors Louisiana Entertainment Commission (www.louisianaentertainment.gov), PMC, Ltd. (www.pmc-speakers.com) and Korus (www.korussound.com); Bronze Sponsors GC Pro (stores.guitarcenter.com/gcpro), Lurssen Mastering (www.lurssenmastering.com), Sound Exchange (www.soundexchange.com), Vintage King Audio (www.vintageking.com); and Participating Sponsors iZotope (www.izotope.com), Capital Bauer Studio Insurance Program (www.ajgrms.com/albany), Prism Sound (www.prismsound.com) and Full Sail University (www.fullsail.edu).
These entities join the previously announced Platinum Sponsor AUPEO! (www.aupeo.com) and Gold Sponsors AVID (www.avid.com), Westlake Pro, (www.westlakepro.com), DTS (www.dts.com), Shure (www.shure.com), Iron Mountain Entertainment Services (www.ironmountain.com), Music Marketing (www.musicmarketing.ca), and Presonus (www.presonus.com).
"We are pleased to announce these additional sponsors of the P&E Wing GRAMMY Week Celebration, our annual event that has become a highlight of this very special week in music," said Maureen Droney, Senior Executive Director of the P&E Wing. "Our honoree this year, Neil Young, has long been a passionate spokesperson for better sound quality and is himself pioneering new and innovative ways to deliver recorded music that sounds as close as possible to the studio quality that artists hear when they're recording. We are grateful for the participation of the manufacturers and organizations that partner with us for this event, themselves overall supporters of sonic fidelity and the art and craft of recorded music."
Young's career began with his earliest recordings as a member of the Squires in Winnipeg, Canada, in 1963. He rose to prominence as a solo artist with classic albums such as 1970's After The Gold Rush and 1972's Harvest, and through work with bands such as Buffalo Springfield (Buffalo Springfield, Buffalo Springfield Again); Crosby, Stills, Nash & Young (American Dream, Déjà Vu); and Neil Young With Crazy Horse (Everybody Knows This Is Nowhere). He has also directed and released films through his production company Shakey Pictures, including Rust Never Sleeps (1979) and the documentary CSNY/Déjà Vu (2008). His most recent release, Neil Young With Crazy Horse's Psychedelic Pill,shows an artist fully grounded and firmly rooted in his work yet constantly evolving and living in the moment.
Young continues to push artistic boundaries by releasing Neil Young Archives Vol. 1 1963–1972 in a variety of formats, including Blu-ray, thus allowing fans to continue to receive new Neil Young archival content as it becomes available. He has received numerous awards and accolades throughout his career including an Oscar nomination in '93 for his song "Philadelphia" from the film Philadelphia. He was inducted into the Canadian Music Hall of Fame in '82, and has twice been inducted into the Rock and Roll Hall of Fame — in '95 for his solo work, and in '97 as a member of Buffalo Springfield. In 2014, he will debut the highly anticipated Pono, his new high fidelity audio system. Young's philanthropic work is as renowned as his musical legacy. He was one of the original founders of Farm Aid in 1985, and continues to be an active member on the organization's board of directors. Each October, he and his wife Pegi organize and host the Bridge School Concerts, a benefit for the Bridge School that he and Pegi helped found. The school ensures that children with severe speech and physical impairments achieve full participation in their communities through the use of assistive technology and the development, implementation and dissemination of innovative educational strategies. These weekend concerts have drawn an international array of musical artists and have raised significant resources for the school. For more than four decades, Young has set a standard of artistic integrity and iconoclastic creativity and his achievements have been matched by his unwavering humanitarianism. Neil Young is a shining example of how music people offer their creative gifts to the world, and how they also give back through their commitments to charitable endeavors.


text:
A more equitable climate transition is needed, according to the head of Germany's consumer protection organizations
Home » A more equitable climate transition is needed, according to the head of Germany's consumer protection organizations
The chairman of the German Consumer Organizations Federation (VZBV) has advocated for a climate-friendly infrastructure model that simultaneously ensures a certain quality of living for residents. Klaus Müller remarked in an interview with the daily Der Tagesspiegel that, in addition to regulating industry and pricing, Germany should be developing "climate-compatible infrastructure that guarantees inhabitants, for example, mobility — not only in major cities, but also in the countryside."
"The German government has subsidized the new e-vehicle industry with billions of euros, but it has ignored the used automobile market."
"The most important question is whether I can be confident that a used battery has been independently evaluated and is worth the money when I buy it," he added. He also proposed that reparation payments made through the social security system be tailored to minimize societal divide. Those who travel by bus and train and live in well-insulated homes, for example, should be compensated more than those who drive an SUV and fly frequently.
He also chastised the current government for using electricity prices to fund its industrial policies, resulting in consumers paying the highest electricity prices in Europe while grid operators' "pockets are being lined."
In Germany, the question of who should pay for the climate transition has sparked heated debate. The present CO2 pricing on heating and transportation fuels has been criticized as being unjust to low-income households and has become a major campaign topic.
See also Many paths to carbon neutrality: Who are the contenders for Merkel's successor?
Tags CO2 Europe German Consumer Organizations Federation Germany
Munich Motor Show (Munich Auto Show) With huge demonstrations and 400,000 tourists, the IAA comes to a close
The German government has made the decision to purchase "climate-friendly" goods in the future
Which type of renewable power could speed the energy transition and change our view on clean energy sources?
German transport minister opposes autobahn speed restriction after study reveals benefits for the environment
The main obstacle to getting a commercial location in Bavaria is power grid delays
Portugal will hold its first offshore wind auction, with a goal of 10 GW by 2030


text:
in Apple TV+, News
Apple Debuts Tom Hanks WWII Flim 'Greyhound', 'Little Voice', And 'Greatness Code' On TV+
by Jing Moreno July 10, 2020, 11:20 pm 1.8k Views
Apple Debuts Tom Hanks WWII Flim Greyhound Little Voice And
Ioshackz.com Apple Debuts Tom Hanks WWII Flim 'Greyhound', 'Little Voice', And 'Greatness Code' On TV+ Tom Hanks' WWII film "Greyhound" is now officially available for streaming on Apple TV+.
The film, written by Hanks, focuses on an officer named George Krause, who commands Navy destroyer Greyhound during the Battle of the Atlantic. The film portrays Krause battles with his own self-doubts and personal demons, along with the enemy, to prove that he is in charge.
Greyhound was initially slated for release in the movie theaters on Father's Day weekend, but due to the coronavirus pandemic, Sony Pictures has opened up bids to premier the firm on digital storefronts.
There was a bidding war with multiple streaming services for airing rights, but Apple ultimately secured it, with the company is calculated to have spent approximately $70 million to air the film exclusively on Apple TV+.
Apple TV+ has also debut first four episodes of "Little Voice," a coming of age drama created by J.J. Abrams, Sara Bareilles, and Jessie Nelson, and uses original music by Bareilles. There is also a short-form sports docuseries "Greatness Code" available to stream through the TV app on the iPhone, iPad, iPod touch, Apple TV, Mac, select Samsung and LG smart TVs, Amazon Fire TV, and Roku devices, as well as online at tv.apple.com.
AppleCodeDebutsFlimGreatnessGreyhoundHanksTomVoiceWWII
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BOSTON — Brad Marchand is used to plying his trade in front of thousands of screaming fans. But this past weekend, the Bruins winger was the center of attention on a very different type of stage.
Marchand was a special guest of the world-renowned Boston Pops on Sunday afternoon, reading "Twas the Night Before Christmas" as part of the orchestra's 2015 Holiday Pops concert.
The reading of the famed Christmas poem evidently was a nerve-wracking experience, even for a celebrity like Marchand, who said he practiced his lines for several days leading up to the concert.
"I met with the conductor (Keith Lockhart) before, and he kind of filled me in on how they were going to play and where I was breaking and everything," he said. "I felt a little more calm and confident, but I was still nervous up there. It's a different scene for me.
Marchand was the second Bruins player to join the Pops on stage in recent years, following former B's forward Shawn Thornton. His biggest takeaway? Performing at Symphony Hall is quite a bit different from playing at TD Garden.
"It was different because I was speaking to the crowd," Marchand said. "In (front of) 17,000 people (at the Garden), I'm one of 40 guys on the ice. The attention is kind of not all on me. And it wasn't all on me up there, but I was speaking, and it would have been easy for me to mess up. I was the only one who could kind of screw things up up there.
Marchand will be back in his element Monday night as the Bruins host the Edmonton Oilers.


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Massachusetts Quilt Documentation Project, MassQuilts
Massachusetts Quilt Documentation Project (MassQuilts) initiated in 1994, has documented over 7,000 pre-1950s quilts originating or currently owned in this state. By encouraging the study and preservation of quilts, MassQuilts seeks to demonstrate that these textiles convey important insights about their historical context as well as the lives of the individual women who made them. Monthly documentation days held at the New England Quilt Museum continue to offer the public an opportunity to learn more about their antique quilts.
In 2009, MassQuilts published Massachusetts Quilts: Our Common Wealth, a 336-page hardbound book edited by Lynne Z. Bassett. It presents the most beautiful and historically significant quilts studied in the previous fifteen years of the MassQuilts Documentation Project and includes more than 200 color photos: overall views, detail shots, and photographs of the makers. To see some of the photos and read related stories included in the book visit Sample Essays.
The records of the Massachusetts Quilt Documentation Project-MassQuilts are housed in the New England Quilt Museum.
--Martha Supnik, 11/3/2014.
Fifty-seven Massachusetts museum collections are included in MassQuilts. Click on the name below to see their collections.
New England Quilt Museum
Amherst Historical Society and Museum
Athol Historical Society
Belchertown Historical Association at Stone House Museum
Berkshire Historical Society at Arrowhead
Bidwell House Museum
Boston Museum of Fine Arts
Boylston Historical Society
Braintree Historical Society
Brookline Historical Society
Centerville Historical Museum
Chatham Historical Society, Inc.
Clara Barton Birthplace Museum
Cohasset Historical Society
Concord Museum
Dennis Historical Society
Duxbury Rural Historical Society
Falmouth Historical Society
Granby Historical Association
Hatfield Historical Society
Hingham Historical Society
Historic Deerfield-Memorial Hall Museum
Historic Newton
Historic Northampton
Historical Society of the Town of Hampden, Inc.
Ipswich Museum
Keep Homestead Museum
Louisa May Alcott's Orchard House
Lynn Museum and Historical Society
Martha's Vineyard Historical Society
Mattapoisett Historic Society
Monson Historical Society
Nantucket Historical Association
National Heritage Museum
Nye Family of America Association, Inc.
Osterville Historical Museum
Pelham Historical Society
Pocumtuck Valley Memorial Association at Deerfield
Porter Phelps Huntington Foundation
Ramapoque Historical Society at Josiah Day House
Rehoboth Antiquarium Society, Carpenter Museum
Rowe Historical Society
Royall House Association
Shelburne Historical Society
Sudbury Historical Society, Inc.
Swift River Valley Historical Society
Townsend Historical Society, Reed House
Western Hampden Historical
Westfield Athenaeum
Worcester Historical Museum
Lowell, Massachusetts, United States
The New England Quilt Museum, founded in 1987 and located in the historic district of Lowell, Massachusetts, owns over 500 antique and contemporary quilts, quilt tops, and quilt-related objects.
Amherst, Massachusetts, United States
Athol, Massachusetts, United States
Belchertown, Massachusetts, United States
Pittsfield, Massachusetts, United States
Berkshire, Massachusetts, United States
Monterey, Massachusetts, United States
Boylston, Massachusetts, United States
Braintree, Massachusetts, United States
Brookline, Massachusetts, United States
Gloucester, Massachusetts, United States
Centerville, Massachusetts, United States
Chatham, Massachusetts, United States
Oxford, Massachusetts, United States
Cohasset, Massachusetts, United States
Concord, Massachusetts, United States
Dennis, Massachusetts, United States
Falmouth, Massachusetts, United States
Granby, Massachusetts, United States
Hatfield, Massachusetts, United States
Historical Society, Museum
Hinghham, Massachusetts, United States
Deerfield, Massachusetts, United States
Haverhill, Massachusetts, United States
Newton, Massachusetts, United States
Northampton, Massachusetts, United States
Hampden, Massachusetts, United States
Ipswich, Massachusetts, United States
Monson, Massachusetts, United States
Lynn, Massachusetts, United States
Vineyard Haven, Massachusetts, United States
Mattappoisett, Massachusetts, United States
North Andover, Massachusetts, United States
Nantucket, Massachusetts, United States
Lexington, Massachusetts, United States
East Sandwich, Massachusetts, United States
Sturbridge, Massachusetts, United States
Osterville, Massachusetts, United States
Salem, Massachusetts, United States
Pelham, Massachusetts, United States
Hadley, Massachusetts, United States
West Springfield, Massachusetts, United States
Rehoboth, Massachusetts, United States
Rowe, Massachusetts, United States
Medford, Massachusetts, United States
Shelburne, Massachusetts, United States
Sudbury, Massachusetts, United States
New Salem, Massachusetts, United States
Townsend, Massachusetts, United States
Wenham, Massachusetts, United States
Westfield, Massachusetts, United States
Worcester, Massachusetts, United States


text:
Viewpoint: India and Pakistan up the ante on disputed border
By Ajai Shukla Delhi
The fragile peace along the de facto border dividing India and Pakistan in Kashmir is crumbling.
The Line of Control (LoC) is an un-demarcated 776km (482 mile) stretch of frontier that snakes across the mountains of the northern Indian state of Jammu and Kashmir.
About 200,000 Indian and Pakistani soldiers face each other here, with rifles, machine guns, mortars and artillery at the ready. It is the most intensively manned and militarised border in the world.
Skirmishes between soldiers and artillery duels were a regular feature of the LoC until it fell silent in 2003, when former Indian prime minister, Atal Behari Vajpayee, and former president of Pakistan Pervez Musharraf agreed to an unwritten peace.
That ceasefire has come under growing pressure since 2013, when both sides resumed a deadly game of sneaking across the LoC, killing enemy soldiers and taking body parts for trophies.
India says that Pakistani soldiers violate the ceasefire to facilitate infiltration by separatist Kashmiri and Pakistani militants who cross the LoC in both directions – from the Indian side to get arms and training in Pakistan-controlled Kashmir; and back to the Indian side to fight security forces in the Kashmir valley.
To facilitate this infiltration, Pakistani soldiers open intense fire at Indian posts to keep soldiers' heads down. That provides an opportunity for small groups of militants to thread their way across no-man's land, scale a formidable Indian border fence and then cross through minefields laid between Indian posts.
The Indian Army says Pakistan violated the ceasefire 860 times in 2017, with December alone accounting for 147 violations, the most in any year since the November 2003 ceasefire. Firing has continued in the new year, with January seeing stepped up violations.
Ceasefire violations have doubled over previous years. India says there were 405 violations in 2015 and 449 in 2016.
Pakistan counters that Indian troops violated the ceasefire more than 1,900 times last year. It says there have been more than 75 violations this January alone.
It is hard to apportion responsibility for ceasefire violations.
'Unprovoked fire'
Both sides say their troops encounter "unprovoked fire" which draws an "appropriate response".
Since no impartial observers monitor the LoC in any serious way – the United Nations Military Observer Group in India and Pakistan (UNMOGIP) visits the LoC only occasionally, and never without warning – it is impossible to accurately verify claims.
In a departure from the norm, Indian army chief Bipin Rawat admitted on 12 January that Indian troops were violating the ceasefire to punish Pakistani posts for supporting infiltration by militants.
"Earlier, we were targeting only infiltrating militants [on the LoC]. But these extremists are disposable commodities for Pakistan. Instead, the pain has to be felt by the Pakistan armed forces for supporting infiltration. So we have started targeting (Pakistani) posts and I can assure you that, in these exchanges of fire, he has suffered three-four times the casualties. That is why we get repeated requests from Pakistan to take the ceasefire back to 2003 levels," Gen Rawat told reporters.
Given India's tough stance, there is little prospect of early de-escalation.
In early January, India's army rebuffed a Pakistani request for the two directors general of military operations (DGMO) to meet and discuss de-escalation on the LoC.
While the DGMOs speak every week over a dedicated hotline, a meeting is a rare event directed at solving a serious problem.
The last DGMO meeting took place at Wagah on Christmas Eve 2013, and succeeded in de-escalating a similar rash of ceasefire violations that had flared on the LoC after Pakistani troops killed seven Indian soldiers and mutilated the bodies of two.
This time, however, the Indian army chief has made the return of calm conditional on Pakistan ending support to infiltration. Gen Rawat said: "If we see a drop in infiltration along the LoC we are willing to call for a ceasefire, but not until we see a drop in infiltration levels."
Since Pakistan is unlikely to completely block militant infiltration into Kashmir, low-level conflict could continue on the LoC.
Over decades, both armies have learned how to calibrate violence without letting it escalate into more serious fighting.
Since both sides occupy heavily defended positions, skirmishes invariably end in stalemates with no capture of significant territory by either side – which would be the only realistic trigger for an escalation of fighting.
India and Pakistan have fought three wars and a limited conflict – all but one were over Kashmir. The two countries formalised the ceasefire line as the Line of Control as part of a new resolve to negotiate differences after their 1971 war which ended in Bangladesh splitting from Pakistan.
A full peace, however, has proved elusive.
Ajai Shukla is a Delhi-based columnist on strategic affairs, defence and diplomacy
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Pycroft, James (DNB00)
←Puttenham, George
Pycroft, James
by George Clement Boase
Pycroft, Thomas→
910678Dictionary of National Biography, 1885-1900, Volume 47
Pycroft, JamesGeorge Clement Boase1896
PYCROFT, JAMES (1813–1895), author, second son of Thomas Pycroft of Pickwick, Wiltshire, barrister-at-law, and brother of Sir Thomas Pycroft [q. v.], was born at Geyers House, Wiltshire, in 1813. He matriculated from Trinity College, Oxford, on 25 May 1831, and graduated B.A. in 1836. He was an enthusiastic cricketer, and claimed to have, jointly with Bishop Ryle, instituted the annual Oxford and Cambridge cricket match in 1836 (Oxford Memoirs, ii. 84–210). In the same year he became a student of Lincoln's Inn, but in 1840 abandoned the study of the law, and was ordained in the church of England. At the same time he became second master of the collegiate school at Leicester. He was curate of Chardstock, Dorset, in 1845, and from 1845 to 1856 perpetual curate of St. Mary Magdalen, Barnstaple. He declined further clerical duty, and took up his residence at Bathwick, Bath. Here he devoted his time to literature, and his leisure to cricket, becoming a member of the Lansdown Club. He never obtained much repute as a player, but he was a great authority on the history, rules, and management of the game. He died of influenza at Brighton on 10 March 1895. He had married, on 8 July 1843, Ann, widow of F. P. Alleyn.
In 1859 he published 'Twenty Years in the Church: an Autobiography.' This work, which ran to a fourth edition in 1861, is a religious novel, which was supposed, without much reason, to be a narrative of the writer's own career; a second part, entitled 'Elkerton Rectory,' appeared in 1860, and was reprinted in 1862. His 'Oxford Memoirs: a Retrospect after Fifty Years,' 1886, 2 vols., contains graphic descriptions of the state of the university in his time.
Other books by him are:
'Principles of Scientific Batting,' 1835.
'On School Education, designed to assist Parents in choosing and co-operating with Instructors for their Sons,' Oxford, 1843.
'Greek Grammar Practice,' 1844.
'Latin Grammar Practice,' 1844.
'A Course of English Reading, adapted to every taste and capacity, with Anecdotes of Men of Genius,' 1844; 4th edit. 1861.
'The Collegian's Guide, or Recollections of College Days. Setting forth the Advantages and Temptations of a University Education. By the Rev. * * * * * * * * * *, M.A., —— College, Oxford,' 1845; 2nd edit. 1858.
'Four Lectures on the Advantages of a Classical Education as an Auxiliary to a Commercial Education,' 1847.
'The Cricket Field, or the History and the Science of Cricket,' 1851; 9th edit. 1887.
'Ways and Words of Men of Letters,' 1861.
'Agony Point; or the Groans of Gentility,' 1861, 2 vols.
'The Cricket Tutor,' 1862; a treatise exclusively practical.
'Dragons' Teeth: a Novel,' 1863, 2 vols.
'Cricketana,' 1865.
He also edited Valpy's 'Virgil Improved,' 1846; W. Enfield's 'The Speaker,' 1851; and to Beeton's 'Cricket Book,' by F. Wood, 1866, he contributed 'A Match I was in.'
[Church of England Photographic Portrait Gallery, 1860, pt. xlvii. with portrait; Times, 13 March 1895, p. 10; Wisden's Cricketers' Almanack, 1892, pp. xlix, 1.]
G. C. B.
Retrieved from "https://en.wikisource.org/w/index.php?title=Pycroft,_James_(DNB00)&oldid=3412209"


It blows my mind that Meteora isn't a New World Wonder because it should be. Located in central Greece, this mystical place – a cluster of enormous rock formations topped by monasteries that appear to be suspended in the air almost 1500 feet above sea level – is a stunning masterpiece of nature that's nothing short of a geological and architectural marvel.
Historians believe the rocks got their impressive shapes through drastic erosion during the Middle Ages, and its massive craters, caves, crevices and peaks offered protection to the locals from the raids of several conquerors. Over time, hermits sought refuge there as the remoteness, height and tranquility of the environment helped them feel more in tune with God and divinity. Eventually, they started building monasteries to live in, and 24 structures were completed and occupied by the late 14th century. Only six buildings are in operation today.
The monks practiced their faith with vows of poverty, chastity and obedience, so their lives were simple. And without many material assets, their daily work was hard.
Imagine living a life of fasting and unceasing prayer and having to make that ascent or descent for supplies in baskets, with scaffolding supported on beams that were driven into the rocks! Thankfully, today we can drive on paved roads and walk up or down evenly-placed steps without too much hassle. Even if you have no strong religious leanings, this place is so spectacular and photo op worthy, it is a must-see.
From Athens, you can take a train, car or bus. The travel times are approximately 5 hours by train, 4+ hours by car and 7-8 by bus. You can find details on routes and schedules here.
The more adventurous thrill seekers among us can get all James Bond while there and go rock climbing, exploring caves, mountain biking, rafting, and more. If you need a visual of one type of toe-curling action possible, you need look no further than the movies. Roger Moore actually filmed an action sequence in Meteora at the Monastery of the Holy Trinity for his 1981 Bond movie "For Your Eyes Only". I've included a YouTube clip below. Just don't try it on your own.
Note: To break up the journey to or from Athens, you can stop at the foot of Mount Parnassos to visit the Delphi historic site, a former cultural and religious center which was the sanctuary of Apollo and the shrine of his oracle. It is 80 kilometers northwest of Athens.
These mountains have the power to take your breath away! And I had to be careful about that because this tour involved steep climbs and I was so cold, I was already dealing with the weight of TWO jackets.😉 _ Travel tip: If you're planning a trip to Greece, add Delphi to your itinerary. It's an archeological site – a day trip from Athens – where the 4th century Temple of Apollo existed. History declares it was once home to a legendary oracle who people made long pilgrimages to see for consultation about big decisions. _ As I sat on that bench and marvelled at the detail and scale of the architecture in the compound, I also mused about different customs and practises. Then I looked up and thanked my Father in Heaven for His non-queue access, the gift of travel, and his numerous blessings.


I expect most of you horse gals have either read Black Beauty or seen the movie. It's an amazing book, written by a woman who believed in what she was writing. The movie is quite different from the book, as is often the case.
Anna Sewell wrote Black Beauty, one of the best-loved animal stories ever. This book, written about 130 years ago, changed the way people saw and treated animals, especially horses, forever.
Black Beauty tells the story of a strong and sweet-natured horse and his many owners, of varying levels. It also tells of his friends and a small part of their lives. A lady called Ann Lindo was so shocked and upset by this story that she started a rest home for equines. It's first resident was a overworked cab horse, like Black Beauty himself. This home is still helping equines today. You might have heard of it — it's called The Horse Trust in London, England.
If you've not read or seen Black Beauty yet, I think you should do so as soon as possible.
Editor's Note: The story of Black Beauty, originally published in 1877, is now in the public domain, which means no one owns the original text so it can be freely distributed.
Click here to read the FREE Black Beauty e-book!
If you prefer a hard copy, you should be able to find Black Beauty at your local library.
I loved the movie and the book but I think the book was better. It's definitely my favorite story.
Black Beauty's always been one of my favourite horse books and I think it rocks that Ann Lindo taken the message seriously!
I've read the book and seen the movie they were both great! It's nice that horses have a place to relax at Ann's rest home!
da book n da movie rock! it is 1 of my fav books EVA!
I like the book and the movie a lot. Nice review.
The move was alright, but the book was WAY better!
It is cool because people can read it online!


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BusinessTravels
Home›Business›FlyScoot Customer Service Review
FlyScoot Customer Service Review
Part 1 – Ridiculous Upgrade Fees
Note: This is an on-going review. No conclusions yet. I will update as this situation plays out.
I bought a ticket late August'12 for a ticket to Taipei. Will be going on a few treks with a couple of friends of mine from late December to early January'13. Btw, my ticket to TP is one way because I'll be going to KL on Airasia after that.
So, typically, the earlier you book, the cheaper the fares yea? But surprise surprise, I got an email this morning for discounted rated for the 'ScootBiz' seats. It's the 'Budget Business Class' on scoot, which comes with more seating space, 20kg of check-in luggage allowance and a complimentary meal.
I checked online for the new offer and it was pretty good at $298. I bought mine, which was a 'FlyBag' with 15kg check-in luggage, for $288. $10 more for larger seats, a meal and more check-in space. Why not?
First, I tried the online gateway. Using my booking details to login, I went to search for my wanted flight and…whadya know…all seats are 'Sold Out'.
Yea Right. What a lame way of saying that the seats are reserved for the special offer.
Next, I called them, which connected me to a call centre in Manila. Well, it's a budget airline after all, and it's not in India, so it's fine. Apart from the slightly irritating fake-ish American accents that Filipino call centres seem to have, I've always preferred them to the Indian reps because the Filipinos tend to have better training, attitudes and much more common sense.
Anyway, this is about scoot, not my peeves with outsourced call centres. After confirming my details, I asked if I could upgrade. He was all 'oh sure! let's see…the upgrade will cost you…$98'.
Wait. What? 98 Bucks?
So I told him, ok, I've got the booking screen in front of me and the difference should only be $10. He proceeded to tell me that there was a handling fee of about $70+ and some other fee that's like $8. Can't remember the exact details but I just know that it was ridiculous.
Alright, so I was a little sway, but I was not late or making a last minute change. The flight is 3 months away for crying out loud. Even then, I was actually prepared to pay a small fee for the upgrade. Maybe $10, or $20 even, then if Scoot really want to be a prick, don't include GST and add it on. But paying about 27% of my old ticket price as a HANDLING FEE. What, it takes 27% of the effort used to fly me to TP to upgrade a ticket? You have got to be kidding me.
Most businesses I know reward you for upgrading. Apparently, Scoot punishes early birds and upgraders.
Dear Mr. Campbell Wilson. The purpose of this offer was to encourage people to try out your ScootBiz package right? I'm already a commited customer; I've got a ticket that's paid up in full. Those of us who've booked, shouldn't we be amongst the first to be encouraged to try it out? This early , I'd allow all those who have already booked to get the upgrades and they only need to pay the difference wherever applicable. It's not like other products where it's already been consumed or are being used. Your service has yet to be rendered. A 27% fee is a joke. Again, it's was not like I was doing a last minute change. I wonder what Ron Kaufman would say.
I've been impressed with the setup thus far. A clean, effective and easy to use booking system and from many accounts so far, the flying experience is a-ok. Then this shenanigan.
I've sent an email to Scoot. Let's see how they respond.
Part 2 (In the works, waiting for a response)
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