Superconvergence analysis of the linear finite element method and a gradient recovery postprocessing on anisotropic meshes

For the linear finite element method based on general unstructured anisotropic meshes in two dimensions, we establish the superconvergence in energy norm of the finite element solution to the interpolation of the exact solution for elliptic problems. We also prove the superconvergence of the postprocessing process based on the global L2-projection of the gradient of the finite element solution. Our basic assumptions are: (i) the mesh is quasiuniform under a Riemannian metric and (ii) each adjacent element pair forms an approximate (anisotropic) parallelogram. The analysis follows the same methodology developed by Bank and Xu in 2003 for the case of quasi-uniform meshes, and the results can be considered as an extension of their conclusion to the adaptive anisotropic meshes. Numerical examples involving both internal and boundary layers are presented in support of the theoretical analysis.