Higher order numerical methods for singularly perturbed elliptic problems

Authors:
Justin B. Munyakazi;Kailash C. Patidar
Affiliations:
Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville, South Africa;Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville, South Africa
Venue:
Neural, Parallel & Scientific Computations
Year:
2010

Citing 4
Cited 0

Solution of a boundary value problem for an elliptic equation with a small parameter for the leading derivatives

USSR Computational Mathematics and Mathematical Physics
Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems

Journal of Computational and Applied Mathematics - Special issue: The international conference on computational methods in sciences and engineering 2004
Discontinuous Discretization for Least-Squares Formulation of Singularly Perturbed Reaction-Diffusion Problems in One and Two Dimensions

SIAM Journal on Numerical Analysis
Corrigendum: Corrigendum to "Uniformly convergent finite element methods for singularly perturbed elliptic boundary value problems I: Reaction-diffusion type" [Comput. Math. Appl. 35 (3) (1998) 57-70]

Computers & Mathematics with Applications

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Abstract

We consider a family of singularly perturbed elliptic problems in two dimensions. A novel fitted operator finite difference method developed is proposed to solve this problems. Through a rigorous convergence analysis, we show that the method is second order convergent in both variables. Further attempts are made to improve the order of convergence via some convergence acceleration techniques, namely the Richardson extrapolation. In turn, we achieve fourth order accurate results. Error analysis after extrapolation is also presented. Furthermore, some numerical results confirming the theoretical estimates are provided. We also compare our results with those obtained in the literature (see, e.g., [R. Lin, Discontinuous discretization for least-squares formulation of singularly perturbed reaction-diffusion problems in one and two dimensions, SIAM J. Numer. Anal. 47(1) 89--108.] and noticed that the error obtained by our approach is exponentially smaller than the one obtained by their approach.