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Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^N$ and $u_h$ be a solution of $$ \begin{cases} \partial_t u_h -\Delta_h u_h = f(x) & \text{ in } \Omega_h\\ u_h=0 &\text{ in } \partial \Omega_h \setminus \Omega \end{cases}$$ where $\Delta_h$ is the finite difference approximation of the Laplace operator, $\Omega_h$ is an extension of $\Omega$: $$\Omega_h=\{x \in \mathbb R^N: dist(x,\Omega) < h \}.$$

How can we estimate the error $\Vert u_h - u\Vert_{L^\infty(\bar \Omega)}$, where $u$ is the solution of $$ \begin{cases} \partial_t u -\Delta u = f & \text{ in } \Omega\\ u=0 &\text{ in } \partial \Omega \end{cases}$$ (i.e. the corresponding continuous problem)?

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    $\begingroup$ There are entire books on finite difference methods for PDEs. Also, it is by no means clear what "the" finite difference approximation of the Laplace operator is on a domain that is not a rectangle. $\endgroup$ Mar 3, 2020 at 3:16
  • $\begingroup$ @MichaelRenardy Yes, there are many books on the topic, but I cannot find a rigorous error estimate anywhere. For the definition, we extend the domain and set $u_h = 0$ in $\Omega_h \setminus \Omega$. $\endgroup$
    – Kei
    Mar 3, 2020 at 7:52

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