HIGH-ORDER BOUNDARY-CONDITIONS AND FINITE-ELEMENTS FOR INFINITE DOMAINS

A finite element method for the solution of linear elliptic problems i n infinite domains is proposed. The two-dimensional Laplace, Helmholtz and modified Helmholtz equations outside an obstacle and in a semi-in finite strip, are considered in detail. In the proposed method, an art ificial boundary B is first introduced, to make the computational doma in Omega finite. Then the exact nonlocal Dirichlet-to-Neumann (DtN) bo undary condition is derived on B. This condition is localized, and a s equence of local boundary conditions on B, of increasing order, is obt ained. The problem in Omega, with a localized DtN boundary condition o n B, is then solved using the finite element method. The numerical sta bility of the scheme is discussed. A hierarchy of special conforming f inite elements is developed and used in the layer adjacent to B, in co njunction with the local high-order boundary condition applied on B. A n error analysis is given for both nonlocal and local boundary conditi ons. Numerical experiments are presented to demonstrate the performanc e of the method.