A weak discrete maximum principle and stability of the finite element method in $L_{\infty }$ on plane polygonal domains. I

Abstract:

Let $\Omega$ be a polygonal domain in the plane and $S_r^h(\Omega )$ denote the finite element space of continuous piecewise polynomials of degree $\leqslant r - 1\;(r \geqslant 2)$ defined on a quasi-uniform triangulation of $\Omega$ (with triangles roughly of size h). It is shown that if ${u_h} \in S_r^h(\Omega )$ is a "discrete harmonic function" then an a priori estimate (a weak maximum principle) of the form \[ {\left \| {{u_h}} \right \|_{{L_\infty }(\Omega )}} \leqslant C{\left \| {{u_h}} \right \|_{{L_\infty }(\partial \Omega )}}\] holds. Now let u be a continuous function on $\bar \Omega$ and ${u_h}$ be the usual finite element projection of u into $S_r^h(\Omega )$ (with ${u_h}$ interpolating u at the boundary nodes). It is shown that for any $\chi \in S_r^h(\Omega )$ \[ {\left \| {u - {u_h}} \right \|_{{L_\infty }(\Omega )}} \leqslant c{\left ( {\ln \frac {1}{h}} \right )^{\bar r}}{\left \| {u - \chi } \right \|_{{L_\infty }(\Omega )}},\quad {\text {where}}\;\bar r = \left \{ {\begin {array}{*{20}{c}} 1 & {{\text {if}}\;r = 2,} \\ 0 & {{\text {if}}\;r \geqslant 3.} \\ \end {array} } \right .\] This says that (modulo a logarithm for $r = 2$) the finite element method is bounded in ${L_\infty }$ on plane polygonal domains.