On the implementation of mixed methods as nonconforming methods for second-order elliptic problems

Authors:
Todd Arbogast and Zhangxin Chen

Journal:
Math. Comp. **64** (1995), 943-972

MSC:
Primary 65N30; Secondary 65N22

MathSciNet review:
1303084

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Abstract | References | Similar Articles | Additional Information

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 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 for the known triangular and rectangular elements. The lowest-order Raviart-Thomas mixed solution on rectangular finite elements in and , 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

DOI:
http://dx.doi.org/10.1090/S0025-5718-1995-1303084-8

Keywords:
Finite element,
implementation,
mixed method,
equivalence,
nonconforming method,
multigrid method

Article copyright:
© Copyright 1995
American Mathematical Society