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)?