Towards the implementation of the singular function method for singular perturbation problems

Abstract

The numerical solution of singular perturbation problems (SPPs) is delicate because the perturbation parameter ε and the mesh size h cannot vary independently of one another. In the extended abstract [J.M.-S. Lubuma, K.C. Patidar, Finite element methods for self-adjoint singular perturbation problems, in: T.E. Simos, G. Maroulis (Eds.), ICCMSE 2005: Advances in Computational Methods in Sciences and Engineering, Lecture Series on Computer and Computational Sciences, vol. 4, VSP International Science Publishers, The Netherlands, 2005, pp. 344–347; J.M.-S. Lubuma, K.C. Patidar, Reliable Finite Element Methods for Self-adjoint Singular Perturbation Problems, University of Pretoria Technical Report UPWT 2005/11], the authors proposed the singular function method (SFM) to solve a self-adjoint singular perturbation problem. The SFM is a variant of the finite element method (FEM), where the space of trial and test functions is enriched by the singular functions of the SPP. We proved in the above mentioned work that the SFM is ε-uniformly convergent of optimal order in appropriate norms. However, the numerical implementation of the SFM, based on numerical integration, did not provide satisfactory results. In this paper, we present an alternative way of implementing the SFM. That is, we avoid numerical integration by making use of the explicit form of the singular functions in the computations of the stiffness matrix and the load vector. We obtain improved results. These results are compared with those obtained with numerical integration. The new results confirm theoretical order of convergence.