Optimal Finite Element Mesh Refinement Based on A-posteriori Error Estimator and the Quality of Mesh

A two-level optimal mesh refinement method for h-version finite element analysis of partial differential equations is presented based on both an a-posteriori error indicator and the geometrical quality of mesh. The first level is to refine the meshes on which the a-posteriori error indicators are relatively higher than the others. The error indicators are obtained by simplifying the computation of error bounds which are obtained by solving elemental Neumann type sub problems with the averaged flux for the consistency of the Neumann problems. The simplification of computation means that the functional space on the mesh uniformly refined with only half size of the coarse mesh is chosen as the test functional space in the elemental residual form of error equations, thus the cost for computing the error indicators is very low. After refinement, some refined triangles will become poorly shaped or distorted, then the second level is to move the meshes to improve their geometrical quality with Laplacian smoothing algorithm. A Laplace problem is computed to verify this method and the results show that the refined mesh obtained by both the a-posteriori error indicator and mesh smoothing gives the optimal convergence and more accuracy for the results.