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Error reduction and convergence for an adaptive mixed finite element method

Authors: Carsten Carstensen and R. H. W. Hoppe
Journal: Math. Comp. 75 (2006), 1033-1042
MSC (2000): Primary 65N30, 65N50
Published electronically: March 13, 2006
MathSciNet review: 2219017
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Abstract: An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart-Thomas finite element method with a reduction factor $ \rho<1$ uniformly for the $ L^2$ norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasi-orthogonality estimate. The proof does not rely on duality or on regularity.

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Additional Information

Carsten Carstensen
Affiliation: Department of Mathematics, Humboldt-Universität zu Berlin, D-10099 Berlin, Germany

R. H. W. Hoppe
Affiliation: Institute of Mathematics, Universität Augsburg, D-86159 Augsburg, Germany; and Department of Mathematics, University of Houston, Houston, Texas 77204-3008

Received by editor(s): April 11, 2004
Published electronically: March 13, 2006
Article copyright: © Copyright 2006 American Mathematical Society