Abstract
The boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered. A finite difference scheme on a priori (sequentially) adapted grids is constructed and its convergence is examined. The construction of the scheme on a priori adapted grids is based on a majorant of the singular component of the grid solution that makes it possible to a priori find a subdomain in which the grid solution should be further refined given the perturbation parameter ε, the size of the uniform mesh in x, the desired accuracy of the grid solution, and the prescribed number of iterations K used to refine the solution. In the subdomains where the solution is refined, the grid problems are solved on uniform grids. The error of the solution thus constructed weakly depends on ε. The scheme converges almost ε-uniformly; namely, it converges under the condition N −1 = o(εv), where v = v(K) can be chosen arbitrarily small when K is sufficiently large. If a piecewise uniform grid is used instead of a uniform one at the final Kth iteration, the difference scheme converges ε-uniformly. For this piecewise uniform grid, the ratio of the mesh sizes in x on the parts of the mesh with a constant size (outside the boundary layer and inside it) is considerably less than that for the known ε-uniformly convergent schemes on piecewise uniform grids.
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References
N. S. Bakhvalov, “On the Optimization of Methods for Solving Boundary Value Problems in the Presence of a Boundary Layer,” Zh. Vychisl. Mat. Mat. Fiz. 9, 841–859 (1969).
G. I. Shishkin, Grid Approximations of Singularly Perturbed Elliptic and Parabolic Equations (Ural. Otd. Ross. Akad. Nauk, Yekaterinburg, 1992) [in Russian].
J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems (World Sci., Singapore, 1996).
P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O’Riordan, and G. I. Shishkin, Robust Computational Techniques for Boundary Layers (Chapman & Hall/CRC Press, Boca Raton, FL, 2000).
H.-G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems (Springer, Berlin, 1996).
V. D. Liseikin, Grid Generation Methods (Springer, Berlin, 1999).
P. W. Hemker, G. I. Shishkin, and L. P. Shishkina, “Novel Defect-Correction High-Order, in Space and Time, Accurate Schemes for Parabolic Singularly Perturbed Convection-Diffusion Problems. A Posteriori Adaptive Mesh Technique,” Comput. Meth. Appl. Math. 3, 387–404 (2003).
G. I. Shichkin, “Robust Novel High-Order Accurate Numerical Methods for Singularly Perturbed Convection-Diffusion Problems,” Math. Modelling and Analys. 10, 393–412 (2005).
G. I. Shishkin, “A Posteriori Adaptive (with Respect to the Gradient of the Solution) Grids in Approximation of Singularly Perturbed Convection-Diffusion Equations,” Vychisl. Tekhnol. 6(1–2), 72–87 (2001).
G. I. Shishkin, “Approximation of Singularly Perturbed Reaction-Diffusion Equations on Adapted Grids,” Matem. Modelir. 13(3), 103–118 (2001).
G. I. Shishkin, “The Use of Solutions on Embedded Grids for the Approximation of Singularly Perturbed Parabolic Convection-Diffusion Equations on Adapted Grids,” Zh. Vychisl. Mat. Mat. Fiz. 46, 1617–1637 (2006) [Comput. Math. Math. Phys. 46, 1539–1559 (2006)].
P. W. Hemker, G. I. Shishkin, and L. P. Shishkina, “A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. A Posteriori Adaptive Mesh Technique,” Comput. Meth. Appl. Math. 4(1), 105–127 (2004).
A. A. Samarskii, Theory of Finite Difference Schemes (Nauka, Moscow, 1989; Marcel Dekker, New York, 2001).
A. A. Samarskii and E. S. Nikolaev, Methods for Solving Grid Equations (Nauka, Moscow, 1978) [in Russian].
G. I. Marchuk, Methods of Numerical Mathematics (Nauka, Moscow, 1989; Springer, New York, 1982).
G. I. Marchuk and V. V. Shaidurov, Improving the Accuracy of Finite Difference Schemes (Nauka, Moscow, 1979) [in Russian].
N. S. Bakhvalov, Numerical Methods (Nauka, Moscow, 1973) [in Russian].
P. W. Hemker, G. I. Shishkin, and L. P. Shishikina, “ε-Uniform Schemes with High-Order Time-Accuracy for Parabolic Singular Perturbation Problems,” IMA J. Numer. Analys. 20(1), 99–121 (2000).
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Dedicated to the memory of Academician Aleksandr Andreevich Samarskii
Original Russian Text © G.I. Shishkin, 2008, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2008, Vol. 48, No. 6, pp. 1014–1033.
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Shishkin, G.I. Grid approximation of a parabolic convection-diffusion equation on a priori adapted grids: ε-uniformly convergent schemes. Comput. Math. and Math. Phys. 48, 956–974 (2008). https://doi.org/10.1134/S0965542508060080
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DOI: https://doi.org/10.1134/S0965542508060080