Error reduction and convergence for an adaptive mixed finite element method

Carstensen, Carsten; Hoppe, R.

doi:10.1090/S0025-5718-06-01829-1

Error reduction and convergence for an adaptive mixed finite element method
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by Carsten Carstensen and R. H. W. Hoppe PDF

Math. Comp. 75 (2006), 1033-1042 Request permission

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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