Quasi-optimal convergence rate for adaptive mixed finite element methods

TL;DR

This paper establishes a quasi-optimal convergence rate for adaptive mixed finite element methods by analyzing error decay and data oscillation without restrictive assumptions on the PDE coefficient matrix.

Contribution

It introduces data oscillation analysis without restrictions on the inverse coefficient matrix and proves geometric decay of combined error and estimator, leading to quasi-optimal convergence rates.

Findings

Error in stress variable decays geometrically with adaptive refinement.

Combined error and data oscillation decay rate matches best approximation.

Efficiency of a posteriori error estimator is validated without restrictive assumptions.

Abstract

For adaptive mixed finite element methods (AMFEM), we first introduce the data oscillation to analyze, without the restriction that the inverse of the coefficient matrix of the partial differential equations (PDEs) is a piecewise polynomial matrix, efficiency of the a posteriori error estimator Presented by Carstensen [Math. Comput., 1997, 66: 465-476] for Raviart-Thomas, Brezzi-Douglas-Morini, Brezzi-Douglas-Fortin-Marini elements. Second, we prove that the sum of the stress variable error in a weighted norm and the scaled error estimator is of geometric decay, namely, it reduces with a fixed factor between two successive adaptive loops, up to an oscillation of the right-hand side term of the PDEs. Finally, with the help of this geometric decay, we show that the stress variable error in a weighted norm plus the oscillation of data yields a decay rate in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity