Analysis of the Fourier series Dirichlet-to-Neumann boundary condition of the Helmholtz equation and its application to finite element methods
For researchers in computational wave propagation, this work provides a rigorous analysis of a known boundary condition technique applied to a coupled elastic-acoustic problem, but the contribution is incremental as it extends existing methods to a specific transmission problem.
The authors apply the Fourier series Dirichlet-to-Neumann boundary condition with finite element methods to solve a 2D elastic-acoustic transmission problem, establishing well-posedness and deriving a priori error estimates that account for both discretization and truncation effects. Numerical results demonstrate the accuracy of the scheme.
It is well known that the Fourier series Dirichlet-to-Neumann (DtN) boundary condition can be used to solve the Helmholtz equation in unbounded domains. In this work, applying such DtN boundary condition and using the finite element method, we solve and analyze a two dimensional transmission problem describing elastic waves inside a bounded and closed elastic obstacle and acoustic waves outside it. We are mainly interested in analyzing the DtN boundary condition of the Helmholtz equation in order to establish the well-posedness results of the approximated variational equation, and further derive a priori error estimates involving effects of both the finite element discretization and the DtN boundary condition truncation. Finally, some numerical results are presented to illustrate the accuracy of the numerical scheme.