Regularity and a priori error analysis on anisotropic meshes of a Dirichlet problem in polyhedral domains

Abstract

Consider the Poisson equation on a polyhedral domain with the given data in a weighted $L^2$ space. We establish new regularity results for the solution with possible vertex and edge singularities and propose anisotropic finite element algorithms approximating the singular solution in the optimal convergence rate. In particular, our numerical method involves anisotropic graded meshes with less geometric constraints but lacking the maximum angle condition. Optimal convergence on such meshes usually requires smoother given data. Thus, a by-product of our result is to extend the application of these anisotropic meshes to broader practical computations by allowing almost-$L^2$ data. Numerical tests validate the theoretical analysis.

Appendix

We here illustrate that the numerical convergence rate in (89) is a reasonable indicator for the actual convergence rate. Assume

$$\begin{aligned} |u-u_{j}|_{H^1(\Omega )}=C 2^{-sj}, \end{aligned}$$

where $0<s\le 1$ is the convergence rate and $C>0$ is independent of j. Recall the Galerkin orthogonality

$$\begin{aligned} a(u, u_j)=a(u_j, u_j)\quad {\mathrm{and}}\quad a(u-u_{j-1},u_{j-1})=a(u_j-u_{j-1},u_{j-1})=0. \end{aligned}$$

Then,

$$\begin{aligned}&|u-u_{j}|_{H^1(\Omega )}^2=a(u-u_j, u-u_j)=a(u-u_j, u)=a(u, u)-a(u_j, u_j) \\&\quad =|u|_{H^1(\Omega )}^2-|u_{j}|_{H^1(\Omega )}^2, \end{aligned}$$

and

$$\begin{aligned}&|u_j-u_{j-1}|_{H^1(\Omega )}^2=a(u_j-u_{j-1}, u_j-u_{j-1})=a(u_j-u_{j-1}, u_j) \\&\quad =a(u_j, u_j)-a(u_{j-1}, u_{j-1})=|u_j|_{H^1(\Omega )}^2-|u_{j-1}|_{H^1(\Omega )}^2. \end{aligned}$$

Therefore, we have

$$\begin{aligned} |u_{j}-u_{j-1}|_{H^1(\Omega )}^2= & {} |u_j|_{H^1(\Omega )}^2-|u_{j-1}|_{H^1(\Omega )}^2\\= & {} |u|_{H^1(\Omega )}^2-|u_{j-1}|_{H^1(\Omega )}^2 -\left( |u|_{H^1(\Omega )}^2-|u_{j}|_{H^1(\Omega )}^2\right) \\= & {} |u-u_{j-1}|_{H^1(\Omega )}^2-|u-u_{j}|_{H^1(\Omega )}^2=C^22^{2s(1-j)}-C^22^{-2sj}\\= & {} \left( 1-2^{-2s}\right) C^22^{2s(1-j)}. \end{aligned}$$

This leads to $|u_{j}-u_{j-1}|_{H^1(\Omega )}=\sqrt{1-2^{-2s}}C 2^{s(1-j)}$. Hence,

$$\begin{aligned} \log _2\left( \frac{|u_{j}-u_{j-1}|_{H^1(\Omega )}}{|u_{j+1}-u_{j}|_{H^1(\Omega )}}\right) =s. \end{aligned}$$

In the same manner, one can show that if

$$\begin{aligned} 2^{sj} |u-u_{j}|_{H^1(\Omega )} \rightarrow C \quad {\mathrm{as}}\ j\ {\mathrm{increases}}, \end{aligned}$$

then

$$\begin{aligned} \log _2\left( \frac{|u_{j}-u_{j-1}|_{H^1(\Omega )}}{|u_{j+1}-u_{j}|_{H^1(\Omega )}}\right) \rightarrow s \quad {\mathrm{as}}\ j\ {\mathrm{increases}}. \end{aligned}$$