An Optimal Adaptive Finite Element Method

An optimal adaptive mixed finite element method

Various applications in fluid dynamics and computational continuum mechanics motivate the development of reliable and efficient adaptive algorithms for mixed finite element methods. In order to save degrees of freedom, not all but just a selection of finite element domains are refined. Hence the fundamental question of convergence as well as the question of optimality require new mathematical a...

An Optimal Adaptive Finite Element Method

Preprint No. 1271, Dept. of Math., Univ. of Utrecht, Submitted to SIAM J. Numer. Anal., March 2003 Although existing adaptive finite element methods for solving second order elliptic equations often perform better in practical computations than non-adaptive ones, usually they are even not proven to converge. Only recently in the works of Dörfler ([Dör96]) and that of Morin, Nochetto and Siebert...

An Optimal Adaptive Finite Element Method for the Stokes Problem

A new adaptive finite element method for solving the Stokes equations is developed, which is shown to converge with the best possible rate. The method consists of 3 nested loops. The outermost loop consists of an adaptive finite element method for solving the pressure from the (elliptic) Schur complement system that arises by eliminating the velocity. Each of the arising finite element problems...

Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method

We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As is customary in practice, the AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that the AFEM is a contraction, for the sum of th...

An Adaptive Finite Element Method for Nonlinear Optimal Control Problems