Matrix Representation of a Discretization for a Partial Differential Equation
I want to discretize the following problem
\begin{cases} \mu \nabla^2u+(\lambda+\mu)\nabla \nabla\cdot u = \rho \frac{\partial^2u }{\partial t^2 } + \beta \frac{\partial u}{\partial t}\ u(x,y,t) = c, (x,y) \in \partial( [0,L_{x}]\times [0,L_{y}]), t \in [0,T]\ u(x,y,0) = u_{0}(x,y), (x,y) \in [0,L_{x}]\times [0,L_{y}]\ \frac{\partial u(x,y,0)}{\partial t} = u_{1}(x,y), (x,y) \in [0,L_{x}]\times [0,L_{y}]\ \end{cases}
where $T,L_{x},L_{y} >0$, $c\in \mathbb{R}$ and $$ u_{0},u_{1} \in C^2( [0,L_{x}]\times [0,L_{y}], \mathbb{R^2} ) $$ $$u \in C^2( [0,L_{x}]\times [0,L_{y}]\times [0,T], \mathbb{R^2} ) $$
My attempt. Let $u = ( u_{x} \ u_{y} )^T$
$$ \mu \nabla^2u = ( \mu \nabla^2 u_{x} \ \ \mu \nabla^2u_{y} )^T = $$ $$\mu \begin{pmatrix} \nabla^2 & 0 \ 0 & \nabla^2 \end{pmatrix}\begin{pmatrix} u_{x}\ u_{y} \end{pmatrix} $$
where the laplacian in the <PERSON> product is: $$ \nabla^2 = L_{2}\otimes I_{n_{x}} + I_{n_{y}} \otimes L_{2} $$
\begin{equation} L_{2} = \frac{1}{h^2}\left[\begin{matrix} -2 & 1 & & 0\ 1 & \ddots & \ddots & \ & \ddots & \ddots & 1 \ 0 & & 1 & -2 \end{matrix} \right] \end{equation}
$h$ is the step size or mesh size... Now
$$ (\lambda+\mu)\nabla \nabla \cdot u = $$ $$ \begin{pmatrix} (\lambda+\mu)\frac{\partial^2 u_{x}}{\partial x^2} + (\lambda+\mu)\frac{\partial^2 u_{y}}{\partial x \partial y}\ (\lambda+\mu)\frac{\partial^2 u_{x}}{\partial y \partial x}+(\lambda+\mu)\frac{\partial^2 u_{y}}{\partial y^2} \end{pmatrix} = $$
$$ (\lambda+\mu) \begin{pmatrix} \frac{\partial^2 }{\partial x^2} & \frac{\partial^2 }{\partial x \partial y} \ \frac{\partial^2 }{\partial y \partial x} & \frac{\partial^2 }{\partial y^2} \end{pmatrix} \begin{pmatrix} u_{x} \ u_{y} \end{pmatrix} $$
$$ \frac{\partial^2 }{\partial x^2} = L_{2}\otimes I_{n_{x}} $$ $$ \frac{\partial^2 }{\partial y^2} = I_{n_{y}} \otimes L_{2}$$
My question is how do I represent the following operators in terms of $L_{2}, I, \otimes$ or is not possible ? if it is not possible, what is the matrix representation of...?
$$\frac{\partial^2 }{\partial x \partial y} = \textbf{ ? }$$ $$\frac{\partial^2 }{\partial y \partial x} = \textbf{ ? }$$
Thank you! 935