A Plane Wave Virtual Element Method for the Helmholtz Problem

TL;DR

This paper presents a novel virtual element method combining low order VEM functions and plane waves to efficiently solve the 2D Helmholtz problem with impedance boundary conditions, demonstrating convergence and practical performance.

Contribution

It introduces a new plane wave VEM scheme for the Helmholtz problem, integrating low frequency VEM functions with high-frequency plane waves, and provides convergence analysis and numerical validation.

Findings

Convergence of the proposed method is theoretically established.

Numerical tests show effective performance on polygonal meshes.

The scheme accurately approximates solutions to the Helmholtz problem.

Abstract

We introduce and analyze a virtual element method (VEM) for the Helmholtz problem with approximating spaces made of products of low order VEM functions and plane waves. We restrict ourselves to the 2D Helmholtz equation with impedance boundary conditions on the whole domain boundary. The main ingredients of the plane wave VEM scheme are: i) a low frequency space made of VEM functions, whose basis functions are not explicitly computed in the element interiors; ii) a proper local projection operator onto the high-frequency space, made of plane waves; iii) an approximate stabilization term. A convergence result for the h-version of the method is proved, and numerical results testing its performance on general polygonal meshes are presented.