Domain Decomposition and hp-Adaptive Finite Elements

In this work, we report on an ongoing project to implement an hp-adaptive finite element method. The inspiration of this work came from the development of certain a posteriori error estimates for high order finite elements based on superconvergence Bank and Xu [2003a,b], Bank et al. [2007]. We wanted to create an environment where these estimates could be evaluated in terms of their ability to estimate global errors for a wide range of problems, and to be used as the basis for adaptive enrichment algorithms. Their use in a traditional h-refinement scheme for fixed degree p is straightforward, as is their use for mesh smoothing, again with fixed p. What is less clear and thus more interesting is their use in a traditional adaptive prefinement scheme. One issue we hope to resolve, at least empirically, is the extent to which the superconvergence forming the foundation of these estimates continues to hold on meshes of variable degree. If superconvergence fails to hold globally (for example, in our preliminary experiments, superconvergence seems to hold in the interiors of regions of constant p but fails to hold along interfaces separating elements of different degrees), we would still like to determine if they remain robust enough to form the basis of an adaptive p-refinement algorithm. As this is written, we have implemented in the pltmg package (Bank [2007]) adaptive h-refinement/coarsening, adaptive p-refinement/coarsening, and adaptive mesh smoothing. These three procedures can be used separately, or mixed in arbitrary combinations. For example, one could compose an adap-