Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T07:46:18.412Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE ORDER OF MAXIMUM ERROR OF THE FINITE DIFFERENCE SOLUTIONS OF LAPLACE’S EQUATION ON RECTANGLES

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  01 July 2008

A. A. DOSIYEV  and
S. CIVAL BURANAY
A. A. DOSIYEV*
Affiliation:
Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Cyprus, Mersin 10, Turkey (email: adiguzel.dosiyev@emu.edu.tr)
S. CIVAL BURANAY
Affiliation:
Department of Mathematics, Eastern Mediterranean University, Gazimagusa, Cyprus, Mersin 10, Turkey (email: suzan.buranay@emu.edu.tr)
*
✉For correspondence; e-mail: adiguzel.dosiyev@emu.edu.tr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The finite difference solution of the Dirichlet problem on rectangles when a boundary function is given from C1,1 is analyzed. It is shown that the maximum error for a nine-point approximation is of the order of O(h2(|ln  h|+1)) as a five-point approximation. This order can be improved up to O(h2) when the nine-point approximation in the grids which are a distance h from the boundary is replaced by a five-point approximation (“five and nine”-point scheme). It is also proved that the class of boundary functions C1,1 used to obtain the error estimations essentially cannot be enlarged. We provide numerical experiments to support the analysis made. These results point at the importance of taking the smoothness of the boundary functions into account when choosing the numerical algorithms in applied problems.

Keywords

finite difference methodnonsmooth solutionsuniform error

MSC classification

Secondary: 65N06: Finite difference methods 65N15: Error bounds
Type
Research Article
Information
The ANZIAM Journal , Volume 50 , Issue 1 , July 2008 , pp. 59 - 73
Copyright
Copyright © Australian Mathematical Society 2008

References

[1]Bramble, J. H., Hubbard, B. E. and Thomee, V., “Convergence estimates for essentially positive type Dirichlet problems”, Math. Comp. 23 (1969) 695–709.Google Scholar
[2]Dosiyev, A. A., “A block-grid method of increased accuracy for solving Dirichlet’s problem for Laplace’s equation on polygons”, Comput. Math. Math. Phys. 34 (1994) 591–604.Google Scholar
[3]Dosiyev, A. A., “A fourth order accurate composite grids method for solving Laplace’s boundary value problems with singularities”, Comput. Math. Math. Phys. 42 (2002) 832–849.Google Scholar
[4]Dosiyev, A. A., “On the maximum error in the solution of Laplace equation by finite difference method”, Int. J. Pure Appl. Math. 7 (2003) 229–241.Google Scholar
[5]Dosiyev, A. A., “The high accurate block-grid method for solving Laplace’s boundary value problem with singularities”, SIAM J. Numer. Anal. 42 (2004) 153–178.CrossRefGoogle Scholar
[6]Dosiyev, A. A. and Cival, S., “A combined method for solving Laplace’s boundary value problem with singularities”, Int. J. Pure Appl. Math. 21 (2005) 353–367.Google Scholar
[7]Kantorovich, L. V. and Krylov, V. I., Approximate methods of higher analysis (Noordhoff, Leiden, 1958).Google Scholar
[8]Li, Z. C., Combined methods for elliptic problems with singularities, interfaces and infinities (Kluwer, Dordrecht, 1998).Google Scholar
[9]Samarskii, A. A., The theory of difference schemes (Marcel Dekker, New York, 2001).CrossRefGoogle Scholar
[10]Smith, B. F., Bjorstad, P. E. and Gropp, W. D., Domain decomposition: parallel multilevel methods for elliptic partial differential equations (Cambridge University Press, Cambridge, 1996).Google Scholar
[11]Volkov, E. A., “Differentiability properties of solutions of boundary value problems for the Laplace and Poisson equations on a rectangle”, Proc. Steklov Inst. Math. 77 (1965) 101–126.Google Scholar
[12]Volkov, E. A., “On differential properties of solutions of the Laplace and Poisson equations on a parallelepiped and efficient error estimates of the method of nets”, Proc. Steklov Inst. Math. 105 (1969) 54–78.Google Scholar
[13]Volkov, E. A., “The differential properties of the solutions of Laplace’s equation, and the errors in the method of nets with boundary values in C 2 and C 1,1”, Comput. Math. Math. Phys. 9 (1969) 97–112.CrossRefGoogle Scholar
[14]Volkov, E. A., “On the method of composite meshes for Laplace’s equation on polygons”, Tr. Mat. Inst. Steklova 140 (1976) 68–102.Google Scholar