A finite element method for the two-dimensional Helmholtz equation in unbounded regions

A finite element method for finding approximate solutions of boundary value problems for the Helmholtz equation in two-dimensional unbounded regions is studied. The method is based on a technique developed by R. W. Thatcher for solving the Laplace equation in unbounded regions. . The method uses a grid consisting of an infinite number of triangular elements constructed in a systematic way. A system of linear equations must be solved for the solution at nodes near the boundary. Values at the remaining nodes in the infinite region are calculated from a simple recurrence relation. The method appears very promising but has not yet been tested. Possible extensions are pointed out and discussions of the outward radiation condition and of other methods for solving the Helmholtz equation are included.