Superconvergence Analysis of the Polynomial Preserving Recovery for Elliptic Problems with Robin Boundary Conditions

Abstract

We analyze the superconvergence property of the linear finite element method based on the polynomial preserving recovery (PPR) for Robin boundary elliptic problems on triangulations. First, we improve the convergence rate between the finite element solution and the linear interpolation under the $H^1$-norm by introducing a class of meshes satisfying the $Condition$ $(\alpha,\sigma,\mu)$. Then we prove the superconvergence of the recovered gradients post-processed by PPR and define an asymptotically exact a posteriori error estimator. Finally, numerical tests are provided to verify the theoretical findings.