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

Author: Alfred H. Schatz
Journal: Math. Comp. 34 (1980), 77-91
MSC: Primary 65N30
MathSciNet review: 551291
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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

$\displaystyle {\left\Vert {{u_h}} \right\Vert _{{L_\infty }(\Omega )}} \leqslant C{\left\Vert {{u_h}} \right\Vert _{{L_\infty }(\partial \Omega )}}$


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 )$

$\displaystyle {\left\Vert {u - {u_h}} \right\Vert _{{L_\infty }(\Omega )}} \leq... ...ext{if}}\;r = 2,} \\ 0 & {{\text{if}}\;r \geqslant 3.} \\ \end{array} } \right.$

This says that (modulo a logarithm for $ r = 2$) the finite element method is bounded in $ {L_\infty }$ on plane polygonal domains.

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Article copyright: © Copyright 1980 American Mathematical Society