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Description
Exterior problems in unbounded domains play a central role in the study of wave propagation, particularly in the modeling of scattering phenomena generated by obstacles represented by bounded regions. We focus on a three-dimensional exterior problem for the Helmholtz equation, endowed with Dirichlet boundary conditions on the surface of the obstacle and a suitable radiation condition at infinity.
To approximate its solution, we first introduce an artificial boundary enclosing the obstacle, thereby decomposing the exterior region into a bounded computational domain and an unbounded residual one. We then propose a numerical method based on the coupling of the Virtual Element Method (VEM) for the bounded interior domain (see [1]) with a Boundary Element Method (BEM) defined on the artificial boundary, following the Costabel–Han coupling strategy proposed in [2].
The VEM is by now a well-established and effective framework for problems posed in bounded domains, particularly well suited for handling complex geometries and general polyhedral meshes. Our aim is to extend its advantages to the treatment of unbounded domains through the proposed coupling approach.
To assess the performance and effectiveness of the method, we present a set of numerical experiments.
[1] L. Beirao da Veiga, F. Brezzi, L. Marini, G. Manzini, A. Cangiani, A. Russo (2013), Basic principles of Virtual Element Methods, Math. Models Methods Appl. Sci., 23, 199--214.
[2] G. N. Gatica, S. Meddahi (2020), Coupling of virtual element and boundary element methods for the solution of acoustic scattering problems, J. Numer. Math, 28, 223--245.