Computable two-sided a posteriori error estimates for h-adaptive Finite Element Method

摘要

We describe the simple element-wise a posteriori error estimators (AEEs) that are able to provide two-sided error estimates for Finite Element Method (FEM) approximations of the solutions to the elliptic boundary value problems. We name these estimators "Dirichlet and Neumann estimators" because the theoretical background of their two-sided estimations lays in the analysis of the solutions to the variational formulations of the residual problems with homogeneous Dirichlet and Neumann boundary conditions. Here we show how to construct Dirichlet and Neumann AEEs for the linear finite element approximations on the triangular meshes and prove theoretically their abilities to yield lower and upper bounds of finite element approximation errors for linear problems. The numerical results demonstrate and confirm the properties of Dirichlet and Neumann AEEs in the process of solving singularly perturbed and nonlinear diffusion-advection-reaction problems on the uniform or h-adaptively refined meshes.