Convergence and Optimality of Adaptive Mixed Finite Element Methods

TL;DR

This paper proves the convergence and optimality of adaptive mixed finite element methods for solving the Poisson equation, overcoming challenges due to the lack of a minimization principle.

Contribution

It introduces a novel quasi-orthogonality property and uses discrete stability to establish convergence and optimality of adaptive mixed finite element methods.

Findings

Established convergence of adaptive mixed finite element methods.

Proved optimality using a localized discrete upper bound.

Developed a quasi-orthogonality property for error analysis.

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.