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

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Recent Advances in Computational and Applied Mathematics

Abstract

The hp version of the finite element method (hp-FEM) combined with adaptive mesh refinement is a particularly efficient method for solving partial differential equations because it can achieve a convergence rate that is exponential 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 by p. Several strategies for making this determination have been proposed over the years. In this paper we summarize these strategies and demonstrate the exponential convergence rates with two classic test problems.

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Authors and Affiliations

  1. Mathematical and Computational Sciences Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA

    William F. Mitchell & Marjorie A. McClain

Authors
  1. William F. Mitchell
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  2. Marjorie A. McClain
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Correspondence to William F. Mitchell .

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Editors and Affiliations

  1. Dpt. Computer Science and Technology, University of Peloponnese, Tripolis, GR-221 00, Greece

    Theodore E. Simos

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Mitchell, W.F., McClain, M.A. (2011). A Survey of hp-Adaptive Strategies for Elliptic Partial Differential Equations. In: Simos, T. (eds) Recent Advances in Computational and Applied Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9981-5_10

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