A mixed-type finite element approximation for radiation problems using fictitious domain method

In the finite element approximation of the exterior Helmholtz problem, we propose an approximation method to implement the DtN mapping formulated as a pseudo-differential operator on a computational artificial boundary. The method is then combined with the fictitious domain method. Our method directly gives an approximation matrix for the sesqui-linear form for the DtN mapping. The eigenvalues of the approximation matrix are simplified to a closed form and can be computed efficiently by using a continued fraction formula. Solution outside the computational domain and the far-field solution can also be computed efficiently by expressing them as operations of pseudo-differential operators. An inner artificial DtN boundary condition is also implemented by our method. We prove the convergence of the solution of our method and compare the performance with the standard finite element approximation based on the Fourier series expansion of the DtN operator. The efficiency of our method is demonstrated through numerical examples.