A Comparison of hp -Adaptive Strategies for Elliptic Partial Differential Equations

Abstract

The hp version of the finite element method ( hp -FEM) combined with adaptive mesh refinement is a particularly efficient method for solving PDEs because it can achieve an exponential convergence rate in the number of degrees of freedom. hp -FEM allows for refinement in both the element size, h , and the polynomial degree, p . Like adaptive refinement for the h version of the finite element method, a posteriori error estimates can be used to determine where the mesh needs to be refined, but a single error estimate cannot simultaneously determine whether it is better to do the refinement by h or p . Several strategies for making this determination have been proposed over the years. These strategies are summarized, and the results of a numerical experiment to study the performance of these strategies is presented. It was found that the reference-solution-based methods are very effective, but also considerably more expensive, in terms of computation time, than other approaches. The method based on a priori knowledge is very effective when there are known point singularities. The method based on the decay rate of the expansion coefficients appears to be the best choice as a general strategy across all categories of problems, whereas many of the other strategies perform well in particular situations and are reasonable in general.