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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the implementation of mixed methods as nonconforming methods for second-order elliptic problems
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by Todd Arbogast and Zhangxin Chen PDF
Math. Comp. 64 (1995), 943-972 Request permission

Abstract:

In this paper we show that mixed finite element methods for a fairly general second-order elliptic problem with variable coefficients can be given a nonmixed formulation. (Lower-order terms are treated, so our results apply also to parabolic equations.) We define an approximation method by incorporating some projection operators within a standard Galerkin method, which we call a projection finite element method. It is shown that for a given mixed method, if the projection method’s finite element space ${M_h}$ satisfies three conditions, then the two approximation methods are equivalent. These three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator. We then construct appropriate nonconforming spaces ${M_h}$ for the known triangular and rectangular elements. The lowest-order Raviart-Thomas mixed solution on rectangular finite elements in ${\mathbb {R}^2}$ and ${\mathbb {R}^3}$, on simplices, or on prisms, is then implemented as a nonconforming method modified in a simple and computationally trivial manner. This new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution. A rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal-order multigrid methods for the solution of the linear system.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Math. Comp. 64 (1995), 943-972
  • MSC: Primary 65N30; Secondary 65N22
  • DOI: https://doi.org/10.1090/S0025-5718-1995-1303084-8
  • MathSciNet review: 1303084