A Virtual Element Method coupled with a Boundary Integral Non Reflecting condition for 2D exterior Helmholtz problems

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

“…The very satisfactory results we have obtained in [23] have stimulated us to further investigate on the application of the VEM to the solution of exterior problems. For this reason, we propose here a novel approach in the CVEM-Galerkin context that we have studied both from the theoretical and the numerical point of view.…”

“…We start by describing the quadrature techniques adopted for the computation of the integrals appearing in the local bilinear forms a E h and m E h in (5.6) and (5.7). To this aim we point out that, if the element E is a polygon with straight edges, the choice of M k (E) defined by (5.2) allows for an exact (up to the machine precision) and easy computation of the first term in the right hand side of (5.6) and of (5.7) (for the details we refer to formulas ( 27)-( 30) in [23]). On the contrary, if the element E is a polygon with a curved edge, the corresponding integrals can be computed by applying the n-point Gauss-Lobatto quadrature rule.…”

“…This case corresponds to the matrix entries belonging to the main diagonal and to those co-diagonals, for which the supports of the basis functions overlap or are contiguous. After having introduced the q-smoothing transformation, with q = 3, we have applied the n-point Gauss-Legendre quadrature rule with n = 9 for the outer integrals, and n = 8 for the inner ones (see [24] and Remark 3 in [23] for further details). For the computation of all the other integrals, we have applied a 9 × 8-point Gauss-Legendre product quadrature rule.…”

“…Despite the fact that an integration by parts strategy can be applied to weaken the hypersingularity, the approach turns out to be quite onerous from the computational point of view, especially in the case of frequency-domain wave problems for which the accuracy of the BEM is strictly connected to the frequency parameter and to the density of discretization points per wavelength. Even if the CHC has been applied in several contexts, among which we mention the recent paper [25], where the theoretical analysis has been derived for the solution of the Helmholtz problem by means of a VEM, from the engineering point of view the JNC is the most natural and appealing way to deal with unbounded domain problems (for very recent real-life applications see, for example, [2] and [23]).…”

“…Originally developed as a variational reformulation of the nodal Mimetic Finite Difference (MFD) method [6,15,31], the VEM has been applied to a wide variety of interior problems (among the most recent papers we refer the reader to [3,9,10,16]). On the contrary, only few papers deal with VEM applied to exterior problems, among which the already mentioned [25] and [23]. In this latter the JNC between the collocation BEM and the VEM has been numerically investigated for the approximation of the solution of Dirichlet boundary value problems defined by the 2D Helmholtz equation.…”