Spectral Element Approximations and Infinite Domains

. A spectral-element technique to approximate partial differential equations on an infinite domain is examined. The method is based on Boyd's mapping of a semi-infinite interval to a finite interval, and it is extended to a variational setting which allows for an implementation using a spectral-element method. By extending the method to a variational form, a straight-forward implementation allows for high order approximations over an infinite computational domain. 1. Introduction. Partial Differential Equations (PDEs) on either infinite or semi-infinite domains arise in many applications. For example, for a problem that includes an electromagnetic field in 3-D, the field may need to be extended to an infinite interval to simulate total absorbing boundary conditions. One of the difficulties in dealing with the infinite interval is that these fields decay to zero but only decay algebraically depending on the dimension. The approximation of the resulting equations can give rise to many pr..