On the coupling of the Curved Virtual Element Method with the one-equation Boundary Element Method for 2D exterior Helmholtz problems

TL;DR

This paper introduces a novel coupling of the curved virtual element method with a boundary element method to efficiently solve 2D exterior Helmholtz problems, providing theoretical analysis and numerical validation.

Contribution

It presents a new combined approach for Helmholtz problems involving curved boundaries, with proven optimal convergence and confirmed effectiveness through numerical tests.

Findings

Optimal convergence error estimate in energy norm

Numerical results confirm theoretical predictions

Effective handling of unbounded domains with artificial boundary

Abstract

We consider the Helmholtz equation defined in unbounded domains, external to 2D bounded ones, endowed with a Dirichlet condition on the boundary and the Sommerfeld radiation condition at infinity. To solve it, we reduce the infinite region, in which the solution is defined, to a bounded computational one, delimited by a curved smooth artificial boundary and we impose on this latter a non reflecting condition of boundary integral type. Then, we apply the curved virtual element method in the finite computational domain, combined with the one-equation boundary element method on the artificial boundary. We present the theoretical analysis of the proposed approach and we provide an optimal convergence error estimate in the energy norm. The numerical tests confirm the theoretical results and show the effectiveness of the new proposed approach.