Abstract
While solving the elliptic equations in the canonical domain with the Dirichlet boundary conditions by the grid method, it is obviously, that boundary conditions are satisfied precisely. Therefore it is necessary to expect, that close to the domain boundary the accuracy of the corresponding difference scheme should be higher, than in the middle of the domain. The quantitative estimate of this boundary effect first was announced without proves in 1989 in the Reports of the Bulgarian Academy of sciences by the first author. There accuracy of the difference schemes for two-dimensional elliptic equation with variable coefficients in the divergent form has been investigated.
In this paper ‘weight’ a priori estimates, taking into account boundary effect, for traditional difference schemes, which approximate, with the second order, first boundary problem for quasi-linear elliptic type equation, which main part has a not divergent form, have been obtained.
The paper ends with numerical experiments, which testify to unimprovement, by the order, of the received ‘weight’ estimates.
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References
Makarov, V.: On a priori estimates of difference schemes giving an account of the boundary effect. C.R. Acad. Bulgare Sci. 42(5), 41–44 (1989)
Makarov, V.L., Demkiv, L.I.: Estimates of accuracy of the difference schemes for parabolic type equations taking into account initial-boundary effect. Dopov. Nats. Akad. Nauk. Ukr. 2, 26–32 (2003) (in Ukrainian)
Makarov, V.L., Demkiv, L.I.: Improved accuracy estimates of the difference schemes for parabolic equations. // Praci Ukr. matem. conhresu-2001; Kyiv:Inst. matem. Nats Acad. Nauk. Ukr, pp. 36-47 (2002) (in Ukrainian)
Samarsky, A.A., Lazarov, R.D., Makarov, V.L.: Difference schemes for differential equations with generalized solutions. Vysshaya shkola. Moscow, 296 p. (1987) (in Russian)
Ladyzhenskaya, O.A., Uraltseva, N.N.: Linear and Quasi-linear parabolic-type equations, Nauka, Moscow (1973) (in Russian)
Djakonov, E.G.: Difference methods of solving boundary problems. Moscow Government University, Moscow. part, 1971; part (1972) (in Russian)
Samarskii, A.A.: The theory of the difference scheme. Nauka, Moscow (1977) (in Russian)
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Makarov, V., Demkiv, L. (2005). Accuracy Estimates of Difference Schemes for Quasi-Linear Elliptic Equations with Variable Coefficients Taking into Account Boundary Effect. In: Li, Z., Vulkov, L., Waśniewski, J. (eds) Numerical Analysis and Its Applications. NAA 2004. Lecture Notes in Computer Science, vol 3401. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31852-1_8
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DOI: https://doi.org/10.1007/978-3-540-31852-1_8
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