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Uniform convergence of arbitrary order on nonuniform meshes for a singularly perturbed boundary value problem. (English) Zbl 0856.65089

Summary: We consider the numerical solution of a singularly perturbed linear selfadjoint boundary value problem. Assuming that the coefficients of the differential equation are smooth, we construct and analyze finite difference methods that convergence both with high order and uniformly with respect to the singular perturbation parameter. The analysis is done on a locally quasiuniform mesh, which permits its extension to the case of adaptive meshes which may be used to improve the solution. Numerical examples are presented to demonstrate the effectiveness of the method and its low computational cost. The convergence obtained in practice satisfies the theoretical predictions.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
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