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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2012, Volume 52, Number 9, Pages 1633–1654
(Mi zvmmf9733)
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Uniform grid approximation of nonsmooth solutions with improved convergence for a singularly perturbed convection-diffusion equation with characteristic layers
U. H. Zhemuhov Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119991 Russia
Abstract:
A mixed boundary value problem for a singularly perturbed elliptic convection-diffusion equation with constant coefficients in a square domain is considered. Dirichlet conditions are specified on two sides orthogonal to the flow, and Neumann conditions are set on the other two sides. The right-hand side and the boundary functions are assumed to be sufficiently smooth, which ensures the required smoothness of the desired solution in the domain, except for neighborhoods of the corner points. Only zero-order compatibility conditions are assumed to hold at the corner points. The problem is solved numerically by applying an inhomogeneous monotone difference scheme on a rectangular piecewise uniform Shishkin mesh. The inhomogeneity of the scheme lies in that the approximating difference equations are not identical at different grid nodes but depend on the perturbation parameter. Under the assumptions made, the numerical solution is proved to converge ε-uniformly to the exact solution in a discrete uniform metric at an O(N−3/2ln2N) rate, where N is the number of grid nodes in each coordinate direction.
Key words:
singularly perturbed convection-diffusion equation, mixed boundary value problem, finite-difference method, layer-adapted mesh, characteristic boundary layer, corner singularity, uniform convergence of grid approximations.
Received: 17.10.2011 Revised: 13.03.2012
Citation:
U. H. Zhemuhov, “Uniform grid approximation of nonsmooth solutions with improved convergence for a singularly perturbed convection-diffusion equation with characteristic layers”, Zh. Vychisl. Mat. Mat. Fiz., 52:9 (2012), 1633–1654; Comput. Math. Math. Phys., 52:9 (2012), 1239–1259
Linking options:
https://www.mathnet.ru/eng/zvmmf9733 https://www.mathnet.ru/eng/zvmmf/v52/i9/p1633
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Abstract page: | 260 | Full-text PDF : | 83 | References: | 75 | First page: | 13 |
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