George C. Hsiao

Tao Yin, George C. Hsiao, Liwei Xu

This paper is concerned with boundary integral equation methods for solving the two-dimensional fluid-solid interaction problem. We reduce the problem to three differential systems of boundary integral equations via direct and indirect approaches. Existence and uniqueness results for variational solutions of boundary integral equations equations are established. Since in all these boundary variational formulations, the hypersingular boundary integral operator associated with the time-harmonic Navier equation is a dominated integral operator, we also include a new regularization formulation for this hypersingular operator, which allows us to treat the hypersingular kernel by a wealkly singular kernel. Numerical examples are presented to verify and validate the theoretical results.

George C. Hsiao, Tonatiuh Sanchez-Vizuet, Francisco--Javier Sayas

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 is used for time evolution in the pure BIE formulation and combined with time stepping in the coupled BEM/FEM scenario. Second order convergence in time is proven for BDF2-CQ and numerical experiments are provided for both BDF2 and Trapezoidal Rule CQ showing second order behavior for the latter as well.

Constantin Bacuta, Matthew E. Hassell, George C. Hsiao et al.

This paper proposes and analyzes a full discretization of the exterior transient Stokes problem with Dirichlet boundary conditions. The method is based on a single layer boundary integral representation, using Galerkin semidiscretization in the space variables and multistep Convolution Quadrature in time. Convergence estimates are based on a Laplace domain analysis, which translates into a detailed study of the exterior Brinkman problem. Some numerical experiments are provided.