Convergence of Adaptive Finite Element Methods for Semi-Linear Elliptic Partial Differential Equations

Abstract

We analyze a standard adaptive finite element method (AFEM) for second order semi-linear elliptic partial differential equations with vanishing boundary over a polygonal domain in R^{2}. We prove a contraction property for the weighted sum of the energy error and the error estimator between any two consecutive loops, which implies the convergence of AFEM. The result is obtained based on the assumptions that the initial triangulation is sufficiently refined and a Lipschitz constant is sufficiently small in order to deal with the nonlinear inhomogeneous term f(x, u(x)), which is also assumed to be Lipschitz in the second variable.