A posteriori adaptive mesh technique with a priori error estimates for singularly perturbed semilinear parabolic convection-diffusion equations

Authors:
G. I. Shishkin
Affiliations:
Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, Ekaterinburg 620219, Russia
Venue:
International Journal of Computing Science and Mathematics
Year:
2007

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Abstract

Dirichlet problem is considered for a singularly perturbedsemilinear parabolic convection-diffusion equation on a rectangulardomain. The solution of the classical finite difference scheme on auniform mesh converges at the rate O((ε +N−1)−1 N−1 +N0−) where N + 1 and N0 + 1denote the numbers of mesh points with respect to χ and trespectively, ε ∈ (0, 1] is the perturbationparameter. Using nonlinear and linearised basic classical schemes,finite difference schemes on a posteriori adaptive meshes based onuniform subgrids are constructed. The subdomains where gridrefinement is required are defined by the gradients of thesolutions of the intermediate discrete problems. The constructeddifference schemes converge 'almost ε-uniformly', namely,at the rateO((ε−υN−1) +N−1/2 + N0−1) whereυ is an arbitrary number from (0, 1].