Numerical Solutions For Exterior Problems Of The Helmholtz Equation

Many engineering computation problems can be attributed to the boundary value problems of partial differential equations on unbounded domains. However, for the problem of unbounded domain, the usual methods are finite element method, coupling method, spectral method, natural boundary element method, domain decomposition algorithm and so on. In this paper, the natural boundary element method and the finite element method based on the Dirchlet-to-Neumann (DtN) boundary condition are improved. We study to solve the exterior problem of the Helmholtz equation in two parts as follows:The first part proposes a modified natural boundary element method (MNBEM) by using a modified DtN operator on unbounded domains, proves the existence and uniqueness of solution of the corresponding variational problem in L2(?), and obtains the error estimate of boundary element solution in H1 (?). The proposed method overcomes the ill-posedness brought by the truncation of the integral kernel series in the NBEM which is based on a DtN operator. The numerical results demonstrate the convergence and the advantage of the MNBEM with respect to those of the NBEM.In the second part, we investigate a finite element method with a modified Dirichlet-to-Neumann boundary condition (MDtN-FEM) for the Helmholtz equation on unbounded domains. The a priori error estimates depending on the mesh size, the location of MDtN boundary and the truncation of the series in MDtN are established in the H1- and L2-norms. Numerical examples demonstrate the advantage in accuracy and efficiency for the method.