High-order boundary conditions and finite elements for infinite domains

Abstract

A finite element method for the solution of linear elliptic problems in infinite domains is proposed. The two-dimensional Laplace, Helmholtz and modified Helmholtz equations outside an obstacle and in a semi-infinite strip, are considered in detail. In the proposed method, an artificial boundary B is first introduced, to make the computational domain Ω finite. Then the exact nonlocal Dirichlet-to-Neumann (DtN) boundary condition is derived on B . This condition is localized, and a sequence of local boundary conditions on B , of increasing order, is obtained. The problem in Ω, with a localized DtN boundary condition on B , is then solved using the finite element method. The numerical stability of the scheme is discussed. A hierarchy of special conforming finite elements is developed and used in the layer adjacent to B , in conjunction with the local high-order boundary condition applied on B . An error analysis is given for both nonlocal and local boundary conditions. Numerical experiments are presented to demonstrate the performance of the method.