Adaptive mesh refinement and a posteriori error estimates

Abstract

This short contribution is intended mainly for mathematicians who are not specialists in numerical analysis but would like to understand better the fundamental features of the finite element method. First, we review the finite element method for linear elliptic partial differential equations of second order. Then we concentrate on the main ideas of a priori and a posteriori error estimates, convergence and adaptive mesh refinement. We especially emphasize the pioneering convergence result of Professor Miloš Zlámal and present some modern results from the theory of the finite element method. We use several numerical examples to illustrate the presented results.