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

In: Recent Advances in Computational and Applied Mathematics

Author

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  • William F. Mitchell

    (National Institute of Standards and Technology, Mathematical and Computational Sciences Division)

  • Marjorie A. McClain

    (National Institute of Standards and Technology, Mathematical and Computational Sciences Division)

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.

Suggested Citation

  • William F. Mitchell & Marjorie A. McClain, 2011. "A Survey of hp-Adaptive Strategies for Elliptic Partial Differential Equations," Springer Books, in: Theodore E. Simos (ed.), Recent Advances in Computational and Applied Mathematics, chapter 0, pages 227-258, Springer.
    • Handle: RePEc:spr:sprchp:978-90-481-9981-5_10
    DOI: 10.1007/978-90-481-9981-5_10
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