Adaptive Finite Element Algorithms of Optimal Complexity

Abstract

This thesis is dedicated to the design and analysis of optimal algorithms for solving partial differential equations (PDEs). We are primarily interested in the development of efficient algorithms, with respect to computer memory usage and execution time, for solving numerically PDEs. A new adaptive finite element method for solving the Stokes equations is developed, which is shown to converge with the best possible rate. We have implemented this adaptive algorithm and we present a discussion of the numerical performance of our method for the Stokes problem on an L-shaped domain.