Numerical Methods For Transmission Problems And Transmission Eigenvalue Problems In Acoustics

Time-harmonic acoustic wave transmission problem and transmission eigenvalue problem have wide applications in many practical scientific and engineering fields. Transmission eigenvalues can be used to obtain estimates for the material properties of the scattering object, and are of great importance in the uniqueness proof and reconstruction of solutions in inverse scattering theory. This thesis is devoted to the study of numerical methods for transmission problem and transmission eigenvalue problem in acoustics. This thesis consists of two parts. The first part is the numerical methods for transmission problem in acoustics, and the second part is the numerical methods for transmission eigenvalue problem in acoustics.For the transmission problem in acoustics, among the most conventional numerical methods addressing the infinite domain are boundary integral equation methods and the coupled finite element methods. For example, Hsiao and Xu applied the boundary integral equation methods to treat the computation on infinite domain(Hsiao and Xu, 2011). In order to use finite element method to solve the transmission problem in unbounded domain, a popular way is to decompose the unbounded domain into a bounded domain and an unbounded domain by introducing an artificial boundary enclosing the obstacle inside. Concerning the exterior acoustic scattering problem outside the artificial boundary, we can define two different Dirichlet-to-Neumann(DtN) operators by using Fourier expansion series and boundary integral operators. Then, the original unbounded transmission problem is reduced to an equivalent nonlocal boundary value problem by introducing DtN operators on artificial boundaries. Uniqueness and existence of solutions in appropriate Sobolev spaces are established for the corresponding variational problems.For the transmission eigenvalue problem, in order to avoid computing non self-adjoint eigenvalue problems directly, we transform the transmission problem to a series of equivalent self-adjoint fourth eigenvalue problem. Then, we propose an interior penalty discontinuous Galerkin(C~0 IPG) method using Lagrange elements to study fourth order eigenvalue problems. We are not able to apply the theory of Babu?ka-Osborn directly for the proof of convergence of solutions due to the effect of the lower order on the convergence of norm of discrete operators. To overcome this difficulty, following the abstract convergence theory developed in(Descloux et al, 1978a) and the spirit of DG method for the Laplace eigenvalue prolbem(Antonietti et al, 2006), we first show that the method is spectrally correct and then prove the optimal convergence. For a non self-adjoint fourth order eigenvalue problem transformed from the original transmission eigenvalue problem, we give the C~0 IPG method for the discrete fourth eigenvalue problem, and prove the optimal convergence. For high order elliptic problems, C~0 IPG method is much easier for numerical implementation compared with classical conforming finite element methods since it uses simple basis functions and has less degrees of freedom.As for each proposed numerical method, numerical experiments are also presented to illustrate the efficiency and accuracy of the methods.