On Pointwise Estimates for a Finite Element Solution of Nonlinear Boundary Value Problems

Abstract

The solution of the Dirichlet problem for a class of quasilinear elliptic equations is approximated by the simplest finite element method. Piecewise linear trial functions are used over a regular triangulation of the n-dimensional domain with simplices of diameter at most h. Under conditions which are sufficient for the existence of a classical solution we prove pointwise error estimates. In the uniformly elliptic case and for the minimal surface problem the finite element solution converges uniformly with a rate which depends on n and the shape of the domain. It is, e.g., at least $h^{{3 / 2}} | {\log h} |^{{1 / 2}} $ for $n = 2$ and h for $n = 3$. Numerical results are given for two minimal surfaces.