A priori error estimate for Dirichlet problem under geometric uncertainty

Question

I am no specialist in PDE theory but I am interested where I could find an answer for the following question.

Consider two sets $D_1 \subset D_2 \in \mathbb{R}^d$ (actually only for $d \in \{2,3\}$) and the PDEs $$\Delta u_i = f \text{ on } D_i\\u_i=0 \text{ on } \partial D_i $$ with $f:D_2 \rightarrow \mathbb{R}$. Assume that all the usual conditions on regularity for the occurring functions and boundaries are fulfilled.

Are there some kind of error estimates between the solutions $u_1, u_2$? I am sure there are, but where can I find them?

For the case I haven't been rigorous enough let me put it in words: Are there error estimates for the solution of the Dirichlet problem with homogeneous boundary conditions in case of geometric "uncertainty"?

Thanks!

Answer 1

You can use the maximum principle to get error estimates. Since $w:=u_1-u_2$ is harmonic in $D_1$ we have

$$\max_{D_1} |u_1-u_2| \leq \max_{\partial D_1}|u_1-u_2| = \max_{\partial D_1} |u_2|.$$

Now, to estimate the boundary term, you need some notion of closeness of $D_1$ and $D_2$. For example, let us set

$$\varepsilon = \max\{\text{dist}(x,\partial D_2) \, : \, x \in \partial D_1 \}.$$

Given $D_1,D_2$ are open bounded with sufficiently smooth boundaries, the solution $u_2$ is Lipschitz continuous, and so $|u_2(x)| \leq C\text{dist}(x,\partial D_2)$. Therefore $\max_{\partial D_1} |u_2| \leq C\varepsilon$ and so

$$\max_{D_1} |u_1-u_2| \leq C\varepsilon.$$

There are other conditions you can place on the closeness of $D_1$ and $D_2$. If you want to measure the difference in terms of the measure of $D_2\setminus D_1$, I would try energy methods, though this may be harder. In general the solutions can be much different if the domains are not similar.

Answer 2

You can translate the uncertainty from the geometry to the right hand side.

Let $\phi:D^2\to D^1$ be invertible and sufficiently smooth.

Let $v:=u_1\circ \phi$. Then $$ \Delta v= \Delta u+\epsilon = f+\epsilon \\ v(\partial D_2)=0 $$ Hence $$ \|u_2-v\|\leq C\|\epsilon\| $$ and $$ \|u_1-u_2\|\leq C \|\epsilon\| +K $$ with $\epsilon$ and $K$ depending on $\phi$ and $C$ depending on $D_2$