# Convergence and optimality of adaptive mixed finite element methods

Long Chen, Michael Holst, X. U. Jinchao

Research output: Contribution to journalArticle

55 Citations (Scopus)

### Abstract

The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. a quasi-orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation.

Original language English (US) 35-53 19 Mathematics of Computation 78 265 https://doi.org/10.1090/S0025-5718-08-02155-8 Published - Jan 12 2009

### Fingerprint

Mixed Finite Element Method
Optimality
Divergence-free
Orthogonality
Finite element method
Poisson equation
Poisson's equation
Subspace
Oscillation
Upper bound
Approximation

### All Science Journal Classification (ASJC) codes

• Algebra and Number Theory
• Computational Mathematics
• Applied Mathematics

### Cite this

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Convergence and optimality of adaptive mixed finite element methods. / Chen, Long; Holst, Michael; Jinchao, X. U.

In: Mathematics of Computation, Vol. 78, No. 265, 12.01.2009, p. 35-53.

Research output: Contribution to journalArticle

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AB - The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. a quasi-orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation.

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