Convergence and quasi-optimality of an adaptive finite element method for controlling L2 errors

Abstract. In this paper, a contraction property is proved for an adaptive finite element method for controlling the global L2 error. Furthermore, it is shown that the method converges in L2 with the best possible rate. The method that is analyzed is the standard adaptive method except that, if necessary, additional refinements are made to keep the meshes sufficiently mildly graded. This modification does not compromise the quasi-optimality of the resulting algorithm. Key words. Adaptive finite element methods; convergence and quasi-optimality of adaptive finite element methods. AMS subject classification. 65N15, 65N3