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

Technology 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 can not 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.