Convergence of AFEM for Second Order Semi-linear Elliptic PDEs

Thanatyod Jampawai, Khamron Mekchay

Abstract

We analyzed a standard adaptive finite element method (AFEM) for second order semi-linear elliptic partial differential equations (PDEs) with vanishing boundary over a polyhedral domain in R^d, for d bigger than or equal to 2.

Based on a posteriori error estimates using standard residual technique,

we proved the contraction property for the weighted sum of the energy error and the error estimator between two consecutive iterations, which also leads to the convergence of AFEM.

The obtained result is based on the assumptions that the initial mesh or triangulation is sufficiently refined and the nonlinear inhomogeneous term f(x,u) is Lipschitz in the second variable.