MAX-NORM ERROR ESTIMATES FOR FINITE ELEMENT METHODS FOR NONLINEAR SOBOLEV EQUATIONS

  • CHOU, SO-HSIANG (Department of Mathematics and Statistics Bowling Green State University) ;
  • LI, QIAN (Department of Mathematics Shandong Normal University)
  • Published : 2001.12.31

Abstract

We consider the finite element method applied to nonlinear Sobolev equation with smooth data and demonstrate for arbitrary order ($k{\geq}2$) finite element spaces the optimal rate of convergence in $L_{\infty}\;W^{1,{\infty}}({\Omega})$ and $L_{\infty}(L_{\infty}({\Omega}))$ (quasi-optimal for k = 1). In other words, the nonlinear Sobolev equation can be approximated equally well as its linear counterpart. Furthermore, we also obtain superconvergence results in $L_{\infty}(W^{1,{\infty}}({\Omega}))$ for the difference between the approximate solution and the generalized elliptic projection of the exact solution.