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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 )        \[    ‖ u −  u h ‖ L ∞  ( Ω  ) ⩽  c ( ln ⁡  1 h ) r ¯  ‖ u −  χ  ‖ L ∞  ( Ω  ) , where r ¯  = { 1 a m p ; if r = 2 , 0 a m p ; if r ⩾  3. {\left \| {u - {u_h}} \right \|_{{L_\infty }(\Omega )}} \leqslant c{\left ( {\ln \frac {1}{h}} \right )^{\bar r}}{\left \| {u - \chi } \right \|_{{L_\infty }(\Omega )}},\quad {\text {where}}\;\bar r = \left \{ {\begin {array}{*{20}{c}} 1 &amp; {{\text {if}}\;r = 2,} \\ 0 &amp; {{\text {if}}\;r \geqslant 3.} \\ \end {array} } \right .     \]  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.

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