A comparison between finite element methods and spectral methods as applied to bound state problems

The finite element and spectral methods are applied to two-dimensional bound state problems. A comparison of the spectral method, which requires a global basis set expansion of the wave functions, and the finite element method, which requires no such such expansion, is presented. A procedure is given for formulating the finite element approach and for achieving fast and accurate results. The convergence of the finite element calculations is considered and shown to be well behaved. A discussion of the extension of the finite element method to higher dimensions is also included.