Coupling of Finite Elements and Boundary Elements for Transmission Problems of Elastic Waves in R³

Summary

For three-dimensional transmission problems -- describing the scattering of elastic, time-harmonic waves by different, isotropic, linear elastic bodies -- we present a combined approach with finite elements and boundary elements. The given method is based on a general variational principle which renders all boundary conditions on the interface manifold Γ to be natural and also allows inhomogeneous material for the scatterer given here by a bounded domain Ω₁. Our solution procedure makes use of an integral equation method for the exterior problem and of an energy (variational) method for the interior problem and consists of coupling both methods via the transmission condition on the interface. For a given incident field in the unbounded domain Ω₂, this yields a variational principle for the refracted field u_l in the bounded domain ${\Omega _1} = {\text{I}}{{\text{R}}^3}\backslash {\bar \Omega _2}$ and for the traction t₂ of the scattered field u₂ on the boundary Γ of Ω₁. We give uniqueness, existence and regularity results for the true solution. Our approximate solution procedure consists in solving our variational principle with finite elements in Ω₁ and boundary elements on Γ with the Galerkin method. We show convergence and quasioptimality of the Galerkin scheme in the corresponding energy norm.