Higher-order accurate method for a quasilinear singularly perturbed elliptic convection-diffusion equation

G. I. Shishkin

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 9 (2006) no. 1, pp. 81-108.

Résumé

We consider the Dirichlet problem on a rectangle for a quasilinear singularly perturbed elliptic convection-diffusion equation in the case when the domain has no characteristic parts of its boundary; the higher derivatives of the equation contain a parameter е that takes arbitrary values in the half-interval (0,1]. For a linear problem of this type, the $\varepsilon$-uniform rate of convergence for well-known schemes has not higher than the first order (in the maximum norm). For the boundary value problem under consideration, grid approximations are constructed that converge $\varepsilon$-uniformly at the rate $O(N^{-2}\ln^2N)$, where $N$ specifies the number of mesh points in each variable. The piecewise uniform meshes, condensing in the boundary layer, are used. When the values of the parameter are small as compared to the effective mesh size, we apply the domain decomposition method, which is motivated by “asymptotic constructions”. We use monotone approximations of “auxiliary” subproblems that describe the main terms of asymptotic representations of the solutions inside and outside the vicinity of the regular and the angular boundary layers. The above subproblems are solved sequentially on subdomains using uniform meshes. If the values of the parameter are not sufficiently small (as compared to the effective mesh size), then classical finite difference schemes are employed, where the first derivatives are approximated by central difference derivatives. Note that the computation of solutions of the constructed difference scheme, based on the method of “asymptotic constructions”, is essentially simplified for sufficiently small values of the parameter $\varepsilon$.

Bibliographie

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

[2] Ilin A. M., “Raznostnaya skhema dlya differentsialnogo uravneniya s malym parametrom pri starshei proizvodnoi”, Mat. zametki, 6:2 (1969), 237–248

[3] Dulan E., Miller Dzh., Shilders U., Ravnomernye chislennye metody resheniya zadach s pogranichnym sloem, Mir, M., 1983 | MR

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

[5] Farrell P. A., Hemker P. W., Shishkin G. I., “Discrete approximations for singularly perturbed boundary value problems with parabolic layers”, J. Comput. Maths., 14:1 (1996), 71–97 ; 14:2, 183–194 ; 14:3, 273–290 | MR | Zbl | MR | Zbl | MR | Zbl

[6] Miller J. J. H., O'Riordan E., Shishkin G. I., Fitted numerical methods for singular perturbation problems. Error estimates in the maximum norm for linear problems in one and two dimensions, World Scientific, Singapore, 1996 | MR

[7] Roos H.-G., Stynes M., Tobiska L., Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems, Springer, Berlin, 1996 | MR

[8] Kolmogorov V. L., Shishkin G. I., “Numerical methods for singularly perturbed boundary value problems modeling diffusion processes”, Singular Perturbation Problems in Chemical Physics, ed. J. J. H. Miller, John Willey Sons, New York, 1997, 181–362

[9] Farrell P. A., Hegarty A. F., Miller J. J. H., O'Riordan E., Shishkin G. I., Robust computational techniques for boundary layers, Chapman Hall/CRC Press, Boca Raton, 2000 | MR | Zbl

[10] Liseikin V. D., Layer resolving grids and transformations for singular perturbation problems, VSP, Utrecht, 2001

[11] Allen D. N., Southwell R. V., “Relaxation methods applied to determine the motion, in two dimensions, of viscous fluid past a fixed cylinder”, Quart. J. Mech. and Appl. Math., 8:2 (1955), 129–145 | DOI | MR | Zbl

[12] Marchuk G. I., Shaidurov V. V., Povyshenie tochnosti reshenii raznostnykh skhem, Nauka, M., 1979 | MR

[13] K. Böhmer, H. J. Stetter (eds.), Defect correction methods. Theory and applications, Computing Supplementum, 5, Springer, Vienna, 1984 | MR | Zbl

[14] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989 | MR

[15] Marchuk G. I., Metody vychislitelnoi matematiki, Nauka, M., 1989 | MR

[16] Hemker P. W., Shishkin G. I., Shishkina L. P., “The use of defect correction for the solution of parabolic singular perturbation problems”, Z. Angew. Math. Mech., 77:1 (1997), 59–74 | DOI | MR | Zbl

[17] Hemker P. W., Shishkin G. I., Shishkina L. P., “$\varepsilon$-uniform schemes with high-order time-accuracy for parabolic singular perturbation problems”, IMA J. Numer. Anal., 20:1 (2000), 99–121 | DOI | MR | Zbl

[18] Hemker P. W., Shishkin G. I., Shishkina L. P., “High-order time-accurate schemes for parabolic singular perturbation problems with convection”, Russian J. Numer. Anal. Math. Modelling, 17:1 (2002), 1–24 | MR

[19] Hemker P. W., Shishkin G. I., Shishkina L. P., “High-order time-accurate schemes for parabolic singular perturbation convection-diffusion problems with Robin boundary conditions”, Comput. Methods Appl. Math., 2:1 (2002), 3–25 | MR | Zbl

[20] Hemker P. W., Shishkin G. I., Shishkina L. P., “Novel defect correction high-order, in space and time, accurate schemes for parabolic singularly perturbed convection-diffusion problems”, Comput. Methods in Appl. Math., 3:3 (2003), 387–404 | MR | Zbl

[21] Shishkin G. I., “Povyshenie tochnosti reshenii raznostnykh skhem dlya parabolicheskikh uravnenii s malym parametrom pri starshei proizvodnoi”, Zhurn. vychisl. matem. i mat. fiz., 24:6 (1984), 864–875 | MR | Zbl

[22] Shishkin G. I., “Setochnye approksimatsii dlya singulyarno vozmuschennykh ellipticheskikh uravnenii”, Zhurn. vychisl. matem. i mat. fiz., 38:12 (1998), 1989–2001 | MR | Zbl

[23] Khemker P. V., Shishkin G. I., Shishkina L. P., “Dekompozitsiya metoda Richardsona vysokogo poryadka tochnosti dlya singulyarno vozmuschennogo ellipticheskogo uravneniya reaktsii-diffuzii”, Zhurn. vychisl. matem. i mat. fiz., 44:2 (2004), 329–337 | MR

[24] Shishkin G. I., “Setochnye approksimatsii s uluchshennoi skorostyu skhodimosti dlya singulyarno vozmuschennykh ellipticheskikh uravnenii v oblastyakh s kharakteristicheskimi granitsami”, Sib. zhurn. vychisl. matematiki / RAN. Sib. otd-nie, 5:1 (2002), 71–92 | Zbl

[25] Shishkin G. I., “Grid approximation of improved convergence order for a singularly perturbed elliptic convection-diffusion equation”, Proc. of the Steklov Institute of Mathematics, Suppl. 1, 2003, S184–S202 | MR

[26] Emelyanov K. V., “Raznostnaya skhema dlya trekhmernogo ellipticheskogo uravneniya s malym parametrom pri starshikh proizvodnykh”, Kraevye zadachi dlya uravnenii matematicheskoi fiziki, Sb. statei, UNTs AN SSSR, Sverdlovsk, 1973, 30–42

[27] Han H., Kellogg R. B., “Differentiability properties of solutions of the equation $\varepsilon\Delta u+ru=f(x,y)$ in a square”, SIAM J. Math. Anal., 21:2 (1990), 394–408 | DOI | MR | Zbl

[28] 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

[29] Volkov E. A., “O differentsialnykh svoistvakh reshenii kraevykh zadach dlya uravnenii Laplasa i Puassona na pryamougolnike”, Trudy Mat. in-ta AN SSSR, 77, M., 1965, 89–112 | Zbl

[30] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR