Dispersion Analysis of CIP-FEM for Helmholtz Equation

TL;DR

This paper analyzes the numerical dispersion of CIP-FEM for the Helmholtz equation, deriving an explicit penalty parameter formula that significantly reduces pollution errors and improves solution accuracy at high wave numbers.

Contribution

It provides a dispersion analysis for CIP-FEM and derives an explicit penalty parameter formula that reduces phase difference and pollution error.

Findings

Phase difference reduced from O(k(kh)^{2p}) to O(k(kh)^{2p+2})

Pollution error decreased by two orders in kh

Numerical tests confirm improved accuracy with derived penalty parameter

Abstract

When solving the Helmholtz equation numerically, the accuracy of numerical solution deteriorates as the wave number $k$ increases, known as `pollution effect' which is directly related to the phase difference between the exact and numerical solutions, caused by the numerical dispersion. In this paper, we propose a dispersion analysis for the continuous interior penalty finite element method (CIP-FEM) and derive an explicit formula of the penalty parameter for the $p^{\rm th}$ order CIP-FEM on tensor product (Cartesian) meshes, with which the phase difference is reduced from $\mathcal{O}\big(k(kh)^{2p}\big)$ to $\mathcal{O}\big(k(kh)^{2p+2}\big)$. Extensive numerical tests show that the pollution error of the CIP-FE solution is also reduced by two orders in $kh$ with the same penalty parameter.