ε-Uniformly convergent fitted mesh finite difference methods for general singular perturbation problems

摘要

We consider general singular perturbation problems of the form cεy″(x) + a(x)y′(x) + b(x)y(x) = f(x), x ∈ [0, 1]; y(0) = η0, y(1) = η1 with cε equals to both +ε and −ε, a(x), b(x), f(x) are positive throughout the interval and η0,η1∈R. The indirect methods (those which do not use any acceleration of convergence techniques, e.g., Richardson’s extrapolation or defect correction, etc.) for such problems on a mesh of Shishkin type lead the error as O(n-1lnn) where n denotes the total number of sub-intervals of [0, 1]. In this paper, we systematically describe, a very simple and direct method which reduces the error to O(n-2ln2n). This method is proved to be ε-uniformly convergent with the above error bounds, on a piecewise uniform mesh of Shishkin type. The motivation for using this Shishkin mesh is inspired by the quotation of Stynes [M. Stynes, A jejune heuristic mesh theorem, Comput. Methods Appl. Math. 3 (2003) 488–492]: Miller has moved from [J.J.H. Miller, Construction of a FEM for a singularly perturbed problem in 2 dimensions, in: Numerische Behandlung von Differentialgleichungen, Band 2 (Tagung, Math. Forschungsinst., Oberwolfach, 1975), Internat. Ser. Numer. Math., Birkhaüser, Basel, vol. 31, 1976, pp. 165–169] the question “what scheme should one use on a given mesh?” to [P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Robust Computational Techniques for Boundary Layers, Chapman & Hall/CRC, New York, 2000] “what mesh should one use with a given scheme?” The theoretical estimates have been justified by several numerical examples.