Using the digamma function for basis functions in mesh-free computational methods

Abstract

We examine the utility of a new family of basis functions for use with the Complex Variable Boundary Element Method (CVBEM) and other mesh-free numerical methods for solving partial differential equations. The family of polygamma functions have found use in mathematics since as early as 1730 when James Stirling related the digamma function to the factorial function [1]. Now, we propose using the digamma function, as well as new variants of the digamma function, as basis functions for the CVBEM. This paper discusses technical aspects associated with using the digamma function as a CVBEM basis function. Then, we demonstrate the utility of the proposed basis function by applying it to a mixed boundary value problem of the Laplace type.