How a nonconvergent recovered Hessian works in mesh adaptation

Hessian recovery has been commonly used in mesh adaptation for obtaining the required magnitude and direction information of the solution error. Unfortunately, a recovered Hessian from a linear finite element approximation is nonconvergent in general as the mesh is refined. It has been observed numerically that adaptive meshes based on such a nonconvergent recovered Hessian can nevertheless lead to an optimal error in the finite element approximation. This also explains why Hessian recovery is still widely used despite its nonconvergence. In this paper we develop an error bound for the linear finite element solution of a general boundary value problem under a mild assumption on the closeness of the recovered Hessian to the exact one. Numerical results show that this closeness assumption is satisfied by the recovered Hessian obtained with commonly used Hessian recovery methods. Moreover, it is shown that the finite element error changes gradually with the closeness of the recovered Hessian. This provides an explanation on how a nonconvergent recovered Hessian works in mesh adaptation.Comment: Revised (improved proofs and a better example

A posteriori error estimates of mixed discontinuous Galerkin method for a class of Stokes eigenvalue problems

For a class of Stokes eigenvalue problems including the classical Stokes eigenvalue problem and the magnetohydrodynamic Stokes eigenvalue problem a residual type a posteriori error estimate of the mixed discontinuous Galerkin finite element method using $\mathbb{P}_{k}-\mathbb{P}_{k-1}$ element $(k\geq 1)$ is studied in this paper. The a posteriori error estimators for approximate eigenpairs are given. The reliability and efficiency of the posteriori error estimator for the eigenfunction are proved and the reliability of the estimator for the eigenvalue is also analyzed. The numerical results are provided to confirm the theoretical predictions and indicate that the method considered in this paper can reach the optimal convergence order $O(dof^{\frac{-2k}{d}})$

Can we have superconvergent gradient recovery under adaptive meshes

Abstract. We study adaptive finite element methods for elliptic problems with domain corner singularities. Our model problem is the two-dimensional Poisson equation. Results of this paper are twofold. First, we prove that there exists an adaptive mesh (gauged by a discrete mesh density function) under which the recovered gradient by the polynomial preserving recovery (PPR) is superconvergent. Second, we demonstrate by numerical examples that an adaptive procedure with an a posteriori error estimator based on PPR does produce adaptive meshes that asatisfy our mesh density assumption, and the recovered gradient by PPR is indeed superconvergent in the adaptive process