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A posteriori error analysis for defect correction method for two parameter singular perturbation problems. (English) Zbl 1300.65057

The authors apply the defect correction method to the following two-parameter singularly perturbed two-point boundary value problems (BVPs): \[ \begin{gathered} \varepsilon y'' - \mu (b(x)y)' + c(x)y = f(x), \quad x \in (0,1)\\ y(0) = y_0, \quad y(1) = y_1, \end{gathered} \] where \(0 < \varepsilon \ll 1\) and \(0 < \mu \ll 1\).
It is well-known that the solution of the above BVP exhibits boundary layers of different width at both boundaries. Classical finite difference schemes fail to yield satisfactory numerical approximate solutions on uniform meshes. Here, the authors use the Bakhvalov-Shishkin-type meshes to discretize the domain. Then, they apply the well-known postprocessing technique, a defect-correction method to improve the order of accuracy to two. Basically, first, they solve the BVP by a classical upwind scheme and the obtained solution is used to calculate the defect by using the central difference scheme, then they improve the approximate solution. An error analysis is carried out, numerical experiments are done to show the applicability of the method.

MSC:

65L70 Error bounds for numerical methods for ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
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