A least-squares method for the Helmholtz equation

We investigate the use of least-squares methods to approximate the Helmholt z equation. The basis used in the discrete method consists of solutions of the Helmholtz equation (either consisting of plane waves or Bessel function s) on each element of a finite element grid. Unlike previous methods of thi s type, we do not use polynomial based finite elements. The use of small el ements (and relatively few basis functions per element) allows us to prove convergence theorems for the method and, to some extent, control the condit ioning of the resulting linear system. Numerical results show the efficienc y of the new method and suggest that it may be possible to obtain accurate results with a coarser grid than is usual for standard finite element metho ds. (C) 1999 Elsevier Science S.A. All rights reserved.