Abstract
High accurate difference-analytical method of solving the mixed boundary value problem for Laplace’s equation on graduated polygons (which can have broken sections and be multiply connected) is described and justified. The uniform estimate for the error of the approximate solution is of order O(h 4), where h is the mesh step, for the errors of derivatives of order p, p = 1, 2, ..., in a finite neighbourhood of re-entrant vertices, of order O(h 4/r p−λjj ), where r j is the distance from the current point to the vertex in question, λ j = 1/α j or λ j = 1/2α j depending on the types of boundary conditions, α j π is the value of the angle. The last part of the paper is devoted to illustrate numerical experiments.
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References
Li, Z.C.: Combined Methods for Elliptic problems with Singularities, Interfaces and Infinities. Kluwer Academic Publishers, Dordrech Boston London (1998)
Dosiyev, A.A.: The high accurate block-grid method for solving Laplace’s boundary value problem with singularities. SIAM J. Numer. Anal., 42(1), 153–178 (2004)
Dosiyev, A.A.: A block-grid method for increasing accuracy in the solution of the Laplace equation on polygons. Russian Acad. Sci. Dokl.Math., 45(2), 396–399 (1992)
Dosiyev, A.A.: A block-grid method of increased accuracy for solving Dirichlet’s problem for Laplace’s equation on polygons. Comp. Maths Math. Phys., 34(5), 591–604 (1994)
Volkov, E.A.: An exponentially converging method for solving Laplace’s equation on polygons. Math. USSR Sb., 37(3), 295–325 (1980)
Volkov, E.A.: Block method for solving the Laplace equation and constructing conformal mappings. CRC Press, USA (1994)
Dosiyev, A.A., Buranay Cival, S.: A difference-analytical method for solving Laplace’s boundary value problem with singularities. In: Akca, H., Boucherif, A., Covachev, V. (ed) 2004-Dynamical Systems and Applications. GBS Publishers and Distributers, India (2004)
Dosiyev, A.A., Buranay Cival S.: A combined method for solving Laplace’s boundary value problem with singularities. Inter. Journal of Pure and Appl. Math., 21(3), 353–367 (2005)
Dosiyev, A.A.: A fourth order accurate composite grids method for solving Laplace’s boundary value problems with singularities. Comp. Maths Math. Phys., 42(6), 832–849 (2002)
Volkov, E.A.: Effective error estimates for grid method solutions of boundary-value problems for Laplace’s and Poisson’s equations on rectangle and certain triangles. Tr. Mat. Inst. Akad. Nauk SSSR., 74, 55–85 (1966)
Dosiyev, A.A.: On the maximum error in the solution of Laplace equation by finite difference method. Intern. Journal of Pure and Appl. Math., 7(2), 229–241 (2003)
Fix, G.J., Gulati, S., Wakoff, G.I.: On the use of singular functions with finite element approximations. J.Comput. Phys., 13, 209–228 (1973)
Wigley, N.M.: An efficient method for subtracting off singularities at corners for Laplace’s equation. J. Comput. Phys., 78, 369–377 (1988)
Olson, L.G., Georgiou, G.C., Schults, W.W.: An efficient finite element method for treating singularities in Laplace’s equation. J. Comput. Phys., 96, 391–410 (1991)
Dosiyev, A.A., Buranay Cival, S.: On solving the cracked beam problem by a block method. In: Georgiou, G., Papannastasiou, P., and Papadrakakis, M. (ed) 5th GRACM International Congress on Computational Mechanics. Kantzilaris Publication, Nicosia (2005)
Dosiyev, A.A.: A high accuracy difference-analytical method for solving Laplace’s boundary value problem with singularities. Proceedings of the International Conference on Computational Mathematics, Part II, Novosibirsk, 402–407 (2002)
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Dosiyev, A.A., Cival Buranay, S. (2007). A fourth order accurate difference-analytical method for solving Laplace’s boundary value problem with singularities. In: Taş, K., Tenreiro Machado, J.A., Baleanu, D. (eds) Mathematical Methods in Engineering. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5678-9_13
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DOI: https://doi.org/10.1007/978-1-4020-5678-9_13
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