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Optimal error estimate of upwind scheme on Shishkin-type meshes for singularly perturbed parabolic problems with discontinuous convection coefficients. (English) Zbl 1230.65104

A class of singularly perturbed parabolic convection-diffusion problems with discontinuous convection coefficient is considered. Due to the discontinuity of this coefficient the solution possesses interior layers. The analysis of an implicit upwind finite difference scheme on both piecewise-uniform Shishkin mesh and the Bakhalov-Shishkin mesh, for the considered problem, is performed. The authors derive suitable conditions for the appropriate mesh-generating functions which are sufficient for the convergence of the method, uniformly with respect to the perturbation parameter. It is proved from a theoretical and from a numerical point of view as well that the scheme converges uniformly in the discrete supremum norm with an optimal error bound.

MSC:

65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35B25 Singular perturbations in context of PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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