Convergence and Optimality of Adaptive Mixed Finite Element Methods

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

Residual-Based A Posteriori Error Estimates for Symmetric Conforming Mixed Finite Elements for Linear Elasticity Problems

A posteriori error estimators for the symmetric mixed finite element methods for linear elasticity problems of Dirichlet and mixed boundary conditions are proposed. Stability and efficiency of the estimators are proved. Finally, we provide numerical examples to verify the theoretical results

Convergence and optimality of an adaptive modified weak Galerkin finite element method

An adaptive modified weak Galerkin method (AmWG) for an elliptic problem is studied in this paper, in addition to its convergence and optimality. The weak Galerkin bilinear form is simplified without the need of the skeletal variable, and the approximation space is chosen as the discontinuous polynomial space as in the discontinuous Galerkin method. Upon a reliable residual-based a posteriori error estimator, an adaptive algorithm is proposed together with its convergence and quasi-optimality proved for the lowest order case. The major tool is to bridge the connection between weak Galerkin method and the Crouzeix-Raviart nonconforming finite element. Unlike the traditional convergence analysis for methods with a discontinuous polynomial approximation space, the convergence of AmWG is penalty parameter free