Adaptive Solvers for Elliptic and Parabolic Partial Differential Equations

In this thesis our primary interest is in developing adaptive solution methods for parabolic and elliptic partial differential equations. The convection-diffusion equation is used as a representative test problem. Investigations are made into adaptive temporal solvers implementing only a few changes to existing software. This includes a comparison of commercial code against some more academic releases. A novel way to select step sizes for an adaptive BDF2 code is introduced. A chapter is included introducing some functional analysis that is required to understand aspects of the finite element method and error estimation. Two error estimators are derived and proofs of their error bounds are covered. A new finite element package is written, implementing a rather interesting error estimator in one dimension to drive a rather standard refinement/coarsening type of adaptivity. This is compared to a commercially available partial differential equation solver and an investigation into the properties of the two inspires the development of a new method designed to very quickly and directly equidistribute the errors between elements. This new method is not really a refinement technique but doesn�t quite fit the traditional description of a moving mesh either. We show that this method is far more effective at equidistribution of errors than a simple moving mesh method and the original simple adaptive method. A simple extension of the new method is proposed that would be a mesh reconstruction method. Finally the new code is extended to solve steady-state problems in two dimensions. The mesh refinement method from one dimension does not offer a simple extension, so the error estimator is used to supply an impression of the local topology of the error on each element. This in turn allows us to develop a new anisotropic refinement algorithm, which is more in tune with the nature of the error on the parent element. Whilst the benefits observed in one dimension are not directly transferred into the two-dimensional case, the obtained meshes seem to better capture the topology of the solution