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

Abstract

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