A unified analysis of quasi-optimal convergence for adaptive mixed finite element methods

TL;DR

This paper provides a unified framework for analyzing convergence and optimality of adaptive mixed finite element methods, introducing new techniques and verifying hypotheses for key element types in various dimensions.

Contribution

It introduces a novel analysis method based on specific norms and projection operators, establishing convergence and optimality conditions for adaptive mixed finite element methods.

Findings

Proves five key hypotheses ensure convergence and optimality.

Applies the analysis to Raviart--Thomas and Brezzi--Douglas--Marini elements.

Validates the approach for Poisson and Stokes problems in 2D and 3D.

Abstract

In this paper, we present a unified analysis of both convergence and optimality of adaptive mixed finite element methods for a class of problems when the finite element spaces and corresponding a posteriori error estimates under consideration satisfy five hypotheses. We prove that these five conditions are sufficient for convergence and optimality of the adaptive algorithms under consideration. The main ingredient for the analysis is a new method to analyze both discrete reliability and quasi-orthogonality. This new method arises from an appropriate and natural choice of the norms for both the discrete displacement and stress spaces, namely, a mesh-dependent discrete $H^1$ norm for the former and a $L^2$ norm for the latter, and a newly defined projection operator from the discrete stress space on the coarser mesh onto the discrete divergence free space on the finer mesh. As applications, we prove these five hypotheses for the Raviart--Thomas and Brezzi--Douglas--Marini elements of the Poisson and Stokes problems in both 2D and 3D.