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

TL;DR

This paper analyzes the superconvergence properties of DG-FEM with polynomial preserving recovery for the Helmholtz equation, providing explicit error estimates and confirming results through numerical experiments.

Contribution

It establishes superconvergence results for DG-FEM with PPR for high wave number Helmholtz problems, including error estimates and a posteriori error estimators.

Findings

Superconvergence between finite element solution and exact solution under certain conditions.

Superconvergence of recovered gradient via PPR.

Numerical tests confirm theoretical superconvergence results.

Abstract

We study superconvergence property of the linear discontinuous Galerkin finite element method with the polynomial preserving recovery (PPR) and Richardson extrapolation for the two dimensional Helmholtz equation. The error estimate with explicit dependence on the wave number $k$, the penalty parameter $\mu$ and the mesh condition parameter $\alpha$ is derived. First, we prove that under the assumption $k(kh)^2\leq C_0$ ($h$ is the mesh size) and certain mesh condition, the estimate between the finite element solution and the linear interpolation of the exact solution is superconvergent under the $\norme{\cdot}$-seminorm. Second, we prove a superconvergence result for the recovered gradient by PPR. Furthermore, we estimate the error between the finite element gradient and recovered gradient, which motivate us to define the a posteriori error estimator. Finally, Some numerical examples are provided to confirm the theoretical results of superconvergence analysis. All theoretical findings are verified by numerical tests.