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We propose time-domain boundary integral and coupled boundary integral and variational formulations for acoustic scattering by linearly elastic obstacles. Well-posedness along with stability and error bounds with explicit time dependence are established. Full discretization is achieved coupling boundary and finite elements; convolution quadrature (CQ) is used for time evolution in the pure boundary integral equation formulation and combined with time stepping in the coupled boundary-finite element method (BEM-FEM) scenario. Second-order convergence in time is proved for BDF2-CQ and numerical experiments are provided for both BDF2 and trapezoidal rule CQ showing second-order behaviour for the latter as well. [ABSTRACT FROM AUTHOR]

In this manuscript we consider the three dimensional exterior Stokes problem and study the solvability of the corresponding continuous and discrete formulations that arise from the coupling of a dual-mixed variational formulation (in which the velocity, the pressure and the stress are the original main unknowns) with the boundary integral equation method. The present work is an extended and completed version of the analysis and results provided in our previous paper [ZAMM Z. Angew. Math. Mech. 93 (2013), no. 6-7, 437(445). More precisely, after employing the incompressibility condition to eliminate the pressure, we consider the resulting velocity-stress-vorticity approach with different kind of boundary conditions on an annular bounded domain, and couple the underlying equations with either one or two boundary integral equations arising from the application of the usual and normal traces to the Green representation formula in the exterior unbounded region. As a result, we obtain saddle point operator equations, which are then analyzed by the well-known Babuška-Brezzi theory. We prove the well-posedness of the continuous formulations, identifying previously the space of solutions of the associated homogeneous problem, and specify explicit hypotheses to be satisfied by the finite element and boundary element subspaces in order to guarantee the stability of the respective Galerkin schemes. In particular, following a similar analysis given recently for the Laplacian, we are able to extend the classical Johnson & Nédélec procedure to the present case, without assuming any restrictive smoothness requirement on the coupling boundary, but only Lipschitz-continuity. In addition, and differently from known approaches for the elasticity problem, we are also able to extend the Costabel & Han coupling procedure to the 3D Stokes problem by providing a direct proof of the required coerciveness property, that is without argueing by contradiction, and by using the natural norm of each space instead of mesh-dependent norms. Finally, we briefly describe concrete examples of discrete spaces satisfying the aforementioned hypotheses. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we are concerned with the error analysis for the finite element solution of the two-dimensional exterior Neumann boundary value problem in acoustics. In particular, we establish explicit priori error estimates in and - norms including both the effect of the truncation of the DtN mapping and that of the numerical discretization. To apply the finite element method (FEM) to the exterior problem, the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. We discuss essential features of the corresponding variational equation and its modification due to the truncation of the DtN mapping in appropriate function spaces. Numerical tests are presented to validate our theoretical results. [Copyright &y& Elsevier]

Abstract: The interior transmission problem (ITP) is a boundary value problem arising in inverse scattering theory, and it has important applications in qualitative methods. In this paper, we propose a coupled boundary element method (BEM) and a finite element method (FEM) for the ITP in two dimensions. The coupling procedure is realized by applying the direct boundary integral equation method to define the so-called Dirichlet-to-Neumann (DtN) mappings. We show the existence of the solution to the ITP for the anisotropic medium. Numerical results are provided to illustrate the accuracy of the coupling method. [Copyright &y& Elsevier]

Abstract In the finite element and boundary element coupling method, the Stekhlov–Poincaré mapping is often adopted at the coupling boundary, i.e., the unknown Neumann data are represented by the unknown Dirichlet data on the same boundary. This coupling method (either by a convolution integral, or by a Fourier series) is referred as the conventional method. We propose an alternative method, where the unknown Dirichlet data at the boundary are represented by the unknown solution value on an interior curve or surface, i.e., a Dirichlet-to-Dirichlet (D-to-D) mapping is employed instead of a Dirichlet-to-Neumann (D-to-N) mapping. The convergence of the method is shown to be of optimal order. 2D and 3D numerical experiments are performed to demonstrate the accuracy of the proposed method, comparing to the two conventional D-to-N coupling methods. [ABSTRACT FROM AUTHOR]