We consider the Dirichlet problem for an elliptic equation with variable coefficients, the solution of which is obtained by means of a finite-difference scheme of second order accuracy. We establish a two-stage finite-difference method for the posed problem and obtain an estimate of the convergence rate consistent with the smoothness of the solution. It is proved that the solution of the corrected scheme converges at rate in the discrete L2-norm, when the solution of the original problem belongs to the Sobolev space with exponent .
Funding source: Shota Rustaveli National Science Foundation
Award Identifier / Grant number: FR/406/5-106/12
Funding statement: The work was supported by the Shota Rustaveli National Science Foundation under the grant FR/406/5-106/12.
The authors thank an anonymous referee for valuable comments.
1 G. K. Berikelashvili, On the convergence in W22 of the difference solution of the Dirichlet problem (in Russian), Zh. Vychisl. Mat. i Mat. Fiz. 30 (1990), 3, 470–474; translation in USSR Comput. Math. and Math. Phys. 30 (1990), no. 2, 89–92. 10.1016/0041-5553(90)90082-4Search in Google Scholar
2 G. Berikelashvili, The difference schemes of high order accuracy for elliptic equations with lower derivatives, Proc. A. Razmadze Math. Inst. 117 (1998), 1–6. Search in Google Scholar
3 G. Berikelashvili, Construction and analysis of difference schemes for some elliptic problems, and consistent estimates of the rate of convergence, Mem. Differ. Equ. Math. Phys. 38 (2006), 1–131. Search in Google Scholar
4 G. Berikelashvili, M. M. Gupta and M. Mirianashvili, Convergence of fourth order compact difference schemes for three-dimensional convection-diffusion equations, SIAM J. Numer. Anal. 45 (2007), 1, 443–455. 10.1137/050622833Search in Google Scholar
5 J. H. Bramble and S. R. Hilbert, Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math. 16 (1970/1971), 362–369. 10.1007/BF02165007Search in Google Scholar
6 T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces, Math. Comp. 34 (1980), 150, 441–463. 10.1090/S0025-5718-1980-0559195-7Search in Google Scholar
7 L. Fox, Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations, Proc. R. Soc. Lond. Ser. A 190 (1947), 31–59. 10.1098/rspa.1947.0060Search in Google Scholar
8 B. S. Jovanović, The Finite Difference Method for Boundary-Value Problems with Weak Solutions, Posebna Izdan. 16, Matematički Institut u Beogradu, Belgrade, 1993. Search in Google Scholar
9 R. D. Lazarov, V. L. Makarov and A. A. Samarskiĭ, Application of exact difference schemes for constructing and investigating difference schemes on generalized solutions (in Russian), Mat. Sb. (N.S.) 117(159) (1982), 4, 469–480. Search in Google Scholar
10 R. D. Lazarov, V. L. Makarov and W. Weinelt, On the convergence of difference schemes for the approximation of solutions () of elliptic equations with mixed derivatives, Numer. Math. 44 (1984), 2, 223–232. 10.1007/BF01410107Search in Google Scholar
11 A. A. Samarskii, R. D. Lazarov and V. L. Makarov, Difference Schemes for Differential Equations with Generalized Solutions (in Russian), Visshaja Shkola, Moscow, 1987. Search in Google Scholar
12 E. A. Volkov, On a method of increasing the accuracy of the method of grids (in Russian), Dokl. Akad. Nauk SSSR (N.S.) 96 (1954), 685–688. Search in Google Scholar
13 E. A. Volkov, Solving the Dirichlet problem by a method of corrections with higher order differences. I (in Russian), Differ. Uravn. 1 (1965), 7, 946–960. Search in Google Scholar
14 E. A. Volkov, Solving the Dirichlet problem by a method of corrections with higher order differences. II (in Russian), Differ. Uravn. 1 (1965), 8, 1070–1084. Search in Google Scholar
15 E. A. Volkov, On a two-stage difference method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped (in Russian), Zh. Vychisl. Mat. Mat. Fiz. 49 (2009), 3, 512–517; translation in Comput. Math. Math. Phys. 49 (2009), no. 3, 496–501. 10.1134/S0965542509030117Search in Google Scholar
© 2016 by De Gruyter