Conservative finite difference scheme for a singularly perturbed elliptic reaction-diffusion equation: Approximation of solutions and derivatives |
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Authors: | G. I. Shishkin L. P. Shishkina |
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Affiliation: | 1.Institute of Mathematics and Mechanics, Ural Division,Russian Academy of Sciences,Yekaterinburg,Russia |
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Abstract: | A boundary value problem for a singularly perturbed elliptic reaction-diffusion equation in a vertical strip is considered. The derivatives are written in divergent form. The derivatives in the differential equation are multiplied by a perturbation parameter ɛ2, where ɛ takes arbitrary values in the interval (0, 1]. As ɛ → 0, a boundary layer appears in the solution of this problem. Using the integrointerpolational method and the condensing grid technique, conservative finite difference schemes on flux grids are constructed that converge ɛ-uniformly at a rate of O(N 1−2ln2 N 1 + N 2−2), where N 1 + 1 and N 2 + 1 are the number of mesh points on the x 1-axis and the minimal number of mesh points on a unit interval of the x 2-axis respectively. The normalized difference derivatives ɛ k (∂ k /∂x 1 k )u(x) (k = 1, 2), which are ɛ-uniformly bounded and approximate the normalized derivatives in the direction across the boundary layer, and the derivatives along the boundary layer (∂ k /∂ x 2 k )u(x) (k = 1, 2) converge ɛ-uniformly at the same rate. |
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