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Embedding into a rectangular domain

The original domain $\Omega$ is embedded in a rectangle $\Omega^*:=(a_x, b_x)\times(a_y, b_y)$. Parameters $a_x$, $b_x$, $a_y$ and $b_y$ are given in ASSEMBLE/problem_setup.m.

After embedding, the original domain $\Omega$ becomes a `-' phase $\Omega^-$ and the rest is typically denoted by `+' phase $\Omega^+:=\Omega^*\backslash(\overline{\Omega^-})$. The boundary $\partial\Omega$ becomes an interface and boundary conditions turn into jump conditions. The extension is done by zero in $\Omega^+$, thus, on $\partial\Omega^*$ homogeneous Dirichlet boundary conditions can be imposed.

Attention: The embedding rectangle has to be selected in such a way that there is ``enough'' space between $\partial\Omega$ and $\partial\Omega^*$. ``Enough'' depends on the mesh width used in discretisation. In any case, there should be at least several layers of grid points between $\partial\Omega$ and $\partial\Omega^*$.