Finite element solution of nonlinear elliptic problems

Abstract

The study of the finite element approximation to nonlinear second order elliptic boundary value problems with mixed Dirichlet-Neumann boundary conditions is presented. In the discretization variational crimes are commited (approximation of the given domain by a polygonal one, numerical integration). With the assumption that the corresponding operator is strongly monotone and Lipschitz-continuous and that the exact solutionu∈H1(Ω), the convergence of the method is proved; under the additional assumptionu∈H2(Ω), the rate of convergenceO(h) is derived without the use of Green's theorem.