Comparison of hp-Adaptive Finite Element Strategies

Adaptive finite element methods have been studied for nearly 30 years now. Most of the work has focused on h-adaptive methods where the mesh size, h, is adapted locally by means of a local error estimator with the goal of placing the smallest elements in the areas where they will do the most good. h-adaptive methods for elliptic partial differential equations are quite well understood now, and widely used in practice.

Recently, the research community has begun to focus more attention on hp-adaptive methods where in addition to h-adaptivity one locally adapts the degree of the polynomials, p. One attraction of these methods is that they can achieve exponential rates of convergence. But the design of an optimal strategy to determine when to use p-refinement, when to use h-refinement, and what p's to use in h-refined elements is an open area of research. Many such hp-adaptive strategies have been proposed over the past two decades. In this talk, we will briefly describe 13 hp-adaptive strategies and present the results of a numerical experiment to determine which strategies are most effective in terms of error vs. degrees of freedom in different situations.

Speaker Bio: Dr. William F. Mitchell received his Ph.D. in computer science from the University of Illinois in 1988. He has been a computational mathematician in the Applied and Computational Mathematics Division of the National Institute of Standards and Technology for nearly 20 years. His main research interest is the numerical solution of partial differential equations by advanced finite element methods, including adaptive grid refinement, multigrid methods and parallel algorithms.

Presentation Slides: PDF

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