Basis sets and grid techniques

Partial differential equations arise in almost all cases of science, engineering, modelling, and forecasting. Finite differences and finite element methods have long histories as particularly fiexible and powerful general-purpose numerical solution method. In the last two decades, spectral and in particular pseudospectral methods have emerged as intriguing alternatives in many situations - and as superior ones in several areas. For detailed information on 

The finite element method (FEM). The general idea of FEM applied to solve the Schrodinger equation is to change over from the integration to a summation over many subdomains called elements [1, 86]. On each element the wavefunction is approximated by a parametrised function u. The simplest choice are polynomials of different degrees, e.g. in two dimensions 

On each element e a certain number of grid points is choosen and the function u on the element is expanded as 

Integrals over the whole domain of the problem are then sums over all elements. We choose a triangular form of the elements. One obtains a simple integration formula, if one transforms from an arbitrary triangle to a unit rectangular triangle with the coordinates and rj and then to natural triangular coordinates Q (see Fig. la, b in ref. [32])  

The formfunctions can then be expressed in terms of the Q. We have developed general formulas in one and two dimensions and these have been already given [32, 87]. In our calculations we are using regularly polynomials of 5th order, although other choices are possible. 

---