Green element computation of the Sturm-Liouville equations

A mathematical derivation of a new numerical procedure called the Green element method (GEM) is presented and applied to the solution of Sturm-Liouville problems. The GEM is a numerical technique which expands the scope of application of the boundary element method (BEM) by implementing the singular boundary integral theory in an element-by-element fashion; and like the finite element method (FEM) gives rise to a banded coefficient matrix which is easy to handle numerically. For this application, the location of both the field and the source nodes within the same element makes it possible for integrations to be carried out accurately, thereby enhancing the accuracy of discrete equations. The method is therefore easy to apply and, because of its domain based implementation, it maintains the flexibility of the FEM. We apply the GEM to the solution of boundary value differential equations which represent the form of Sturm-Liouville problems, and its capability is demonstrated by comparing the results with those of the finite element methods available in the literature. Satisfactory results and a second-order accuracy were found to be exhibited.