Application of adapted-bubbles to the Helmholtz equation with large wave numbers in 2d

TL;DR

This paper introduces an adapted bubble finite element method for efficiently solving the 2D Helmholtz equation with large wave numbers, significantly reducing pollution errors without needing additional stabilization.

Contribution

The paper proposes a novel adapted bubble approach with a new two-level finite element method for improved accuracy in Helmholtz problems at high wave numbers.

Findings

Effective solution for Helmholtz equation up to ch=3.5

Significant reduction in pollution error

No need for additional stabilization methods

Abstract

An adapted bubble approach which is a modifiation of the residual-free bubbles (RFB) method, is proposed for the Helmhotz problem in 2D. A new two-level finite element method is introduced for the approximations of the bubble functions. Unlike the other equations such as the advection-diffusion equation, RFB method when applied to the Helmholtz equation, does not depend on another stabilized method to obtain approximations to the solutions of the sub-problems. Adapted bubbles (AB) are obtained by a simple modification of the sub-problems. This modification increases the accuracy of the numerical solution impressively. The AB method is able to solve the Helmholtz equation efficiently in 2D up to ch = 3.5 where c is the wave number and h is the mesh size. We provide analysis to show how the AB method mitigates the pollution error.