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

TL;DR

This paper analyzes the superconvergence properties of linear finite element methods combined with polynomial preserving recovery and Richardson extrapolation for the Helmholtz equation, providing error estimates and demonstrating improved accuracy.

Contribution

It establishes superconvergence results for FEM with PPR and Richardson extrapolation for high wave number Helmholtz problems, including error estimates and pollution error cancellation.

Findings

Superconvergence of FEM under certain mesh conditions

PPR improves interpolation error but not pollution error

Richardson extrapolation reduces both interpolation and pollution errors

Abstract

We study superconvergence property of the linear finite element method with the polynomial preserving recovery (PPR) and Richardson extrapolation for the two dimensional Helmholtz equation. The $H^1$-error estimate with explicit dependence on the wave number $k$ {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 $H^1$-seminorm, although the pollution error still exists. Second, we prove a similar result for the recovered gradient by PPR and found that the PPR can only improve the interpolation error and has no effect on the pollution error. Furthermore, we estimate the error between the finite element gradient and recovered gradient and discovered that the pollution error is canceled between these two quantities. Finally, we apply the Richardson extrapolation to recovered gradient and demonstrate numerically that PPR combined with the Richardson extrapolation can reduce the interpolation and pollution errors simultaneously, and therefore, leads to an asymptotically exact {\it a posteriori} error estimator. All theoretical findings are verified by numerical tests.