Geodesic
Uniform grid approximation of nonsmooth solutions with improved convergence for a singularly perturbed convection-diffusion equation with characteristic layers
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 9, pp. 1633-1654 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\varepsilon$-uniformly to the exact solution in a discrete uniform metric at an $O(N^{-3/2}\ln^2N)$ rate, where $N$ is the number of grid nodes in each coordinate direction. 
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     title = {Uniform grid approximation of nonsmooth solutions with improved convergence for a~singularly perturbed convection-diffusion equation with characteristic layers},
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U. H. Zhemuhov. Uniform grid approximation of nonsmooth solutions with improved convergence for a singularly perturbed convection-diffusion equation with characteristic layers. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 9, pp. 1633-1654. http://geodesic.mathdoc.fr/item/ZVMMF_2012_52_9_a5/

[1] Clavero C., Gracia J. L., Lisbona F., Shishkin G. I., “A robust method of improved order for convection-diffusion problems in a domain with characteristic boundaries”, Z. Angew. Math. Mech., 82:9 (2002), 631–647 | 3.0.CO;2-1 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[2] Naughton A., Stynes M., “Regularity and derivative bounds for a convection-diffusion problem with Neumann boundary conditions on characteristic boundaries”, Z. Anal. Anwend., 29:2 (2010), 163–181 | DOI | MR | Zbl

[3] Volkov E. A., “O differentsialnykh svoistvakh reshenii kraevykh zadach dlya uravnenii Laplasa i Puassona na pryamougolnike”, Tr. MIAN SSSR, 77, M., 1965, 89–112 | Zbl

[4] Shih S., Kellogg R. B., “Asymptotic analysis of a singular perturbation problem”, SIAM J. Math. Analys., 18 (1987), 1467–1511 | DOI | MR | Zbl

[5] Shishkin G. I., Setochnye approksimatsii singulyarno vozmuschennykh ellipticheskikh i parabolicheskikh uravnenii, UrO RAN, Ekaterinburg, 1992

[6] Bakhvalov N. S., “K optimizatsii metodov resheniya kraevykh zadach pri nalichii pogranichnogo sloya”, Zh. vychisl. matem. i matem. fiz., 9:4 (1969), 841–859 | Zbl

[7] O'Riordan E., Shishkin G. I., “Parameter uniform numerical methods for singularly perturbed elliptic problems with parabolic boundary layers”, Appl. Numer. Math., 58 (2008), 1761–1772 | DOI | MR

[8] Andreev V. B., “Pointwise approximation of corner singularities for singularly perturbed elliptic problems with characteristic layers”, Internat. J. of Num. Analys. and Modeling, 7:3 (2010), 416–428 | MR

[9] Stynes M., Roos H.-G., “The midpoint upwind scheme”, Appl. Numer. Math., 23:3 (1997), 361–374 | DOI | MR | Zbl

[10] Andreev V. B., Kopteva H. V., “O ravnomernoi po malomu parametru skhodimosti monotonnykh trekh tochechnykh raznostnykh skhem”, Differents. ur-niya, 34:7 (1998), 921–928 | MR | Zbl

[11] Shishkin G. I., “Setochnye approksimatsii s uluchshennoi skorostyu skhodimosti dlya singulyarno vozmuschennykh ellipticheskikh uravnenii v oblastyakh s kharakteristicheskimi granitsami”, Sib. zhurnal vychisl. matem., 5:1 (2002), 71–92 | Zbl

[12] Shishkin G. I., “Setochnaya approksimatsiya metoda dekompozitsii oblasti i resheniya s uluchshennoi skhodimostyu dlya singulyarno vozmuschennykh ellipticheskikh uravnenii v oblastyakh s kharakteristicheskimi granitsami”, Zh. vychisl. matem. i matem. fiz., 45:7 (2005), 1196–1212 | MR | Zbl

[13] Andreev V. B., “Ravnomernaya setochnaya approksimatsiya negladkikh reshenii smeshannoi kraevoi zadachi dlya singulyarno vozmuschennogo uravneniya reaktsii-diffuzii v pryamougolnike”, Zh. vychisl. matem. i matem. fiz., 48:1 (2008), 90–114 | MR | Zbl