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Data structures and requirements for hp finite element software

Published: 16 March 2009 Publication History

Abstract

Finite element methods approximate solutions of partial differential equations by restricting the problem to a finite dimensional function space. In hp adaptive finite element methods, one defines these discrete spaces by choosing different polynomial degrees for the shape functions defined on a locally refined mesh.
Although this basic idea is quite simple, its implementation in algorithms and data structures is challenging. It has apparently not been documented in the literature in its most general form. Rather, most existing implementations appear to be for special combinations of finite elements, or for discontinuous Galerkin methods.
In this article, we discuss generic data structures and algorithms used in the implementation of hp methods for arbitrary elements, and the complications and pitfalls one encounters. As a consequence, we list the information a description of a finite element has to provide to the generic algorithms for it to be used in an hp context. We support our claim that our reference implementation is efficient using numerical examples in two dimensions and three dimensions, and demonstrate that the hp-specific parts of the program do not dominate the total computing time. This reference implementation is also made available as part of the Open Source deal.II finite element library.

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Published In

cover image ACM Transactions on Mathematical Software
ACM Transactions on Mathematical Software  Volume 36, Issue 1
March 2009
137 pages
ISSN:0098-3500
EISSN:1557-7295
DOI:10.1145/1486525
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 16 March 2009
Accepted: 01 August 2008
Revised: 01 April 2008
Received: 01 June 2007
Published in TOMS Volume 36, Issue 1

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Author Tags

  1. hp finite element methods
  2. Object orientation
  3. data structures
  4. finite element software
  5. software design

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  • (2024)An Application-Driven Method for Assembling Numerical Schemes for the Solution of Complex Multiphysics ProblemsApplied System Innovation10.3390/asi70300357:3(35)Online publication date: 24-Apr-2024
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