Analysis of Finite Element Methods and Domain Decomposition Algorithms for a Fluid-Solid Interaction Problem

This paper concerns with the nite element Galerkin approximations for a uid{ solid interaction model proposed in [10]. Both Continuous{time and discrete{time approximations are formulated and analyzed. Optimal order a priori estimates for the errors in L (H) and L(L) are derived. The main di culty for the optimal order error estimates is caused by the interface conditions which describe the interaction between a uid and a solid on their contact surface, and it is overcome by using a boundary duality argument of Douglas and Dupont [5] to handel the terms involving the interface conditions. Finally, several parallelizable domain decomposition algorithms are proposed and analyzed for e ciently solving the nite element systems. x0. Introduction. The problems of wave propagation in composite media have long been subjects of both theoretical and practical studies, important applications of such problems are found in inverse scattering, elastoacoustics, geosciences, oceanography. Di erent mathematical and/or numerical composite models were proposed and studied in [3], [4], [10], [16], [15] and [17]. The purpose of this paper is to analyze the nite element Galerkin approximations for a uid{solid interaction model which was proposed recently by the authors in [10], and to develop some parallelizable domain decomposition algorithms for e ciently solving the nite element systems. In [10] we gave a detailed derivation and the complete mathematical analysis for the model, which will serve as the theoretical foundation for the numerical analysis of this paper. The primary goal of this paper is to establish optimal order a priori error estimates in the L(H){norm and in the L(L){norm for the Galerkin approximations to the solution of the model. The main di culty for obtaining the optimal estimates is caused by the interface conditions which describe the interaction between a uid and a solid on their contact surface, To overcome the di culty, the critical idea is to use a boundary duality argument due to Douglas and Dupont [5] to handel the terms involving the interface conditions. The model and the nite element methods studied in this paper are related to those previously studied by Santos et al [16] and by Sheen [17], where the propagation of waves 1991 Mathematics Subject Classi cation. 65M60, 65M55.