Uniform grid approximation of nonsmooth solutions to the mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a rectangle

V. B. Andreev

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 48 (2008) no. 1, pp. 90-114

Résumé

A mixed boundary value problem for a singularly perturbed reaction-diffusion equation in a square is considered. A Neumann condition is specified on one side of the square, and a Dirichlet condition is set on the other three. It is assumed that the coefficient of the equation, its right-hand side, and the boundary values of the desired solution or its normal derivative on the sides of the square are smooth enough to ensure the required smoothness of the solution in a closed domain outside the neighborhoods of the corner points. No compatibility conditions are assumed to hold at the corner points. Under these assumptions, the desired solution in the entire closed domain is of limited smoothness: it belongs only to the Hölder class $C^\mu$, where $\mu\in(0,1)$ is arbitrary. In the domain, a nonuniform rectangular mesh is introduced that is refined in the boundary domain and depends on a small parameter. The numerical solution to the problem is based on the classical five-point approximation of the equation and a four-point approximation of the Neumann boundary condition. A mesh refinement rule is described under which the approximate solution converges to the exact one uniformly with respect to the small parameter in the $L_\infty^h$ norm. The convergence rate is $O(N^{-2}\ln^2N)$, where $N$ is the number of mesh nodes in each coordinate direction. The parameter-uniform convergence of difference schemes for mixed problems without compatibility conditions at corner points was not previously analyzed.

Bibliographie

[1] Shishkin G. I., Setochnye approksimatsii singulyarno vozmuschennykh ellipticheskikh uravnenii, UO RAN, Ekaterinburg, 1992

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

[3] Andreev V. B., “Setochnye approksimatsii negladkikh reshenii differentsialnykh uravnenii”, Differents. ur-niya, 16:7 (1980), 1172–1184 | MR | Zbl

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

[5] Wigley N. M., “Mixed boundary value problems in plane domains with corners”, Math. Z, 115:1 (1970), 33–52 | DOI | MR | Zbl

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

[7] Volkov E. A., “O differentsialnykh svoistvakh reshenii kraevykh zadach dlya uravneniya Laplasa na mnogougolnikakh”, Tr. MIAN SSSR, 77, M., 1965, 113–142 | Zbl

[8] Andreev V. B., “O tochnosti setochnykh approksimatsii negladkikh reshenii singulyarno vozmuschennogo uravneniya reaktsii-diffuzii v kvadrate”, Differents. ur-niya, 42:7 (2006), 895–906 | MR | Zbl

[9] Clavero C., Gracia J. L., O'Riordan E., “A parameter robust numerical method for a two dimensional recrion-diffusion problem”, Math. Comput., 74 (2005), 1743–1758 | DOI | MR | Zbl

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

[11] Samarskii A. A., Andreev V. B., Raznostnye metody dlya ellipticheskikh uravnenii, Nauka, M., 1976 | MR | Zbl

[12] Volkov E. A., “O metode regulyarnykh sostavnykh setok dlya uravneniya Laplasa na mnogougolnikakh”, Tr. MIAN SSSR, 140, M., 1976, 68–102 | MR | Zbl

[13] M. Abramovits, I. Stigan (red.), Spravochnik po spetsialnym funktsiyam, Nauka, M., 1979 | MR

[14] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, ryadov i proizvedenii, Nauka, M., 1971

[15] Shishmarev I. A., Vvedenie v teoriyu ellipticheskikh uravnenii, Izd-vo MGU, M., 1979 | MR