A Finite Difference Scheme on a Priori Adapted Meshes for a Singularly Perturbed Parabolic Convection-Diffusion Equation |
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Authors: | Grigory I. Shishkin |
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Affiliation: | Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, Ekaterinburg 620219, Russia |
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Abstract: | A boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation; we construct a finite difference scheme on a priori (se-quentially) adapted meshes and study its convergence. The scheme on a priori adapted meshes is constructed using a majorant function for the singular component of the discrete solution, which allows us to find a priori a subdomain where the computed solution requires a further improvement. This subdomain is defined by the perturbation parameter ε, the step-size of a uniform mesh in x, and also by the required accuracy of the discrete solution and the prescribed number of refinement iterations K for im-proving the solution. To solve the discrete problems aimed at the improvement of the solution, we use uniform meshes on the subdomains. The error of the numerical so-lution depends weakly on the parameter ε. The scheme converges almost ε-uniformly, precisely, under the condition N-1 = o(ev), where N denotes the number of nodes in the spatial mesh, and the value v=v(K) can be chosen arbitrarily small for suitable K. |
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Keywords: | Singular perturbations convection-diffusion problem piecewise-uniform mesh a priori adapted mesh |
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