FINITE-ELEMENT APPROXIMATION OF A MODEL REACTION-DIFFUSION PROBLEM WITH A NON-LIPSCHITZ NONLINEARITY

Given lambda > 0 and p member-of(0,1), we consider the following problem: find u such that [GRAPHICS] where OMEGA-subset-of R2 is a smooth convex domain. We prove optimal H-1 and L infinity error bounds for the standard continuous piecewise linear Galerkin finite element approximation. In addition we analyse a more practical approximation using numerical integration on the nonlinear term. Finally we consider a modified nonlinear SOR algorithm, which is shown to be globally convergent, for solving the algebraic system derived from the more practical approximation.