Recent Progress in the Numerical Treatment of Singuiar Problems for Partial Differential Equations With Techniques Based on the Tau Method

Abstract

In this paper we report on recent research work on the use of new formulations of the Tau Method for the numerical solution of singular problems defined by partial differential equations. The problems considered here are relevant to the field of computational fracture mechanics and have attracted considerable attention in the recent literature. Results relate to Laplace's and the biharmonic equations. Comparisons of results obtained by using the Tau Method and other well established methods, such as specificly designed techniques based on the Finite Element Method, the Boundary Integral Method and Collocation, are reported in this paper. These comparisons show that the Tau Method is a versatile technique capable of giving results of a very high accuracy near the singular point with relatively low degrees of approximation. Three formulations of the Tau method are used in this paper for the numerical treatment of singular problems. The first of them, the Tau-Lines approach is a hybrid technique. The second is based on a multidimensional formulation of the Tau Method in terms of Tau elements . In the third one the basis of representation currently used in the Tau Method is augmented with nonpolynomial elements to take into account the singular behaviour of the solution.