A priori error estimates of the DtN-FEM for the transmission problem in acoustics

This paper is concerned with a variational approach solving the two-dimensional acoustic transmission problems. The original problem is reduced to an equivalent nonlocal boundary value problem by introducing an exact Dirichlet-to-Neumann (DtN) mapping in terms of Fourier expansion series on an artificial boundary. Uniqueness and existence of solutions in appropriate Sobolev spaces are established for the corresponding variational problem and its modification due to the truncation of DtN mapping. A priori error estimates containing the effects of both element meshsize and truncation order of series for the finite element approximation are derived. Numerical experiments are also presented to illustrate the efficiency and accuracy of the numerical scheme.