Superconvergence analysis of linear FEM based on polynomial preserving recovery for Helmholtz equation with high wave number

Abstract

Although superconvergence properties of the polynomial preserving recovery (PPR) for finite element methods (FEM) for coercive elliptic problems are well-understood, the superconvergence behavior of PPR for the Helmholtz equation with high wave number still remains unclear. We study the superconvergence property of the linear FEM with PPR for the two dimensional Helmholtz equation on triangulations satisfying the O(h1+α) approximate parallelogram property for some constant α>0. We prove the supercloseness on the H1-seminorm between the finite element solution and the linear interpolant of the exact solution under k(kh)2≤C0, where k is the wave number and h is the mesh size. Then, we obtain the superconvergence result based on the PPR technique which says that the PPR improves only the interpolation error but keeps the pollution error unchanged. Finally, we provide numerical tests to verify the superconvergence property and to demonstrate that the PPR combining with the continuous interior penalty technique is much effective for improving both the interpolation error and the pollution error.