Finite element methods for parameter dependent problems

Abstract

This thesis develops finite element methods for parameter dependent equations. The interest lies in cases where the nature of the problem and the numerical methods used change with the parameters. The studied examples are the reaction-diffusion problem, the Robin boundary condition, which generalizes the Dirichlet and Neumann conditions, and the Brinkman equation, which generalizes the Stokes and the Darcy equations. The developed methods depend continuously on the problem parameters and work even for the limiting values. A posteriori estimates are derived for all the proposed methods, taking into account the parameters. Significant parts of this work are the implementation of the proposed methods and the numerical verfication of the developed theory.