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 Mat. Sb. (N.S.), 1979, Volume 109(151), Number 3(7), Pages 323–354 (Mi msb2387)

An exponentially convergent method for the solution of Laplace's equation on polygons

E. A. Volkov

Abstract: A new approximate method of solving a mixed boundary value problem for Laplace's equation on an arbitrary polygon is presented and substantiated for the case when the right sides in the boundary conditions of the first and second kind on the sides of the polygon are given by algebraic polynomials in the arc length of the boundary of the polygon. By means of this method, an approximate solution of a boundary value problem on a closed polygon can be found with uniform accuracy $\varepsilon>0$ at the expense of $O(|\ln^3\varepsilon|)$ arithmetic operations.
Bibliography: 15 titles.

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English version:
Mathematics of the USSR-Sbornik, 1980, 37:3, 295–325

Bibliographic databases:

UDC: 518.517.944/947
MSC: 35J05, 65N99

Citation: E. A. Volkov, “An exponentially convergent method for the solution of Laplace's equation on polygons”, Mat. Sb. (N.S.), 109(151):3(7) (1979), 323–354; Math. USSR-Sb., 37:3 (1980), 295–325

Citation in format AMSBIB
\Bibitem{Vol79} \by E.~A.~Volkov \paper An exponentially convergent method for the solution of Laplace's equation on polygons \jour Mat. Sb. (N.S.) \yr 1979 \vol 109(151) \issue 3(7) \pages 323--354 \mathnet{http://mi.mathnet.ru/msb2387} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=542804} \zmath{https://zbmath.org/?q=an:0444.35031|0418.35034} \transl \jour Math. USSR-Sb. \yr 1980 \vol 37 \issue 3 \pages 295--325 \crossref{https://doi.org/10.1070/SM1980v037n03ABEH001954} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1980KT31500001} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Volkov YA., “On a Fast Method of Calculation of the Green-Function Corresponding to Laplace Operator on Polygons”, 321, no. 6, 1991, 1143–1146
2. Volkov E., “Rapid Block Method for Constructing Green-Function of Laplace Operator on Polygons”, Differ. Equ., 28:7 (1992), 952–960
3. Dosiyev A., “The Block-Net Method of High-Accuracy Solution of the Laplace Equation of Polygons”, Dokl. Akad. Nauk, 323:4 (1992), 628–631
4. A. A. Dosiev, “A block-grid method of increased accuracy for solving Dirichlet's problem for Laplace's equation on polygons”, Comput. Math. Math. Phys., 34:5 (1994), 591–604
5. Vlasov V. Skorokhodov S., “On Development of Treffitz Method”, Dokl. Akad. Nauk, 337:6 (1994), 713–717
6. Volkov E., “On Solving by Block Method the Laplace Equation on Polygons with Piecewise-Constant Boundary-Value Conditions”, Dokl. Akad. Nauk, 335:5 (1994), 553–555
7. Volkov E., “On a Quick Block Method for Solving the Laplace Equation on Polygons with Nonlocal Boundary-Value Conditions”, Dokl. Akad. Nauk, 342:1 (1995), 11–14
8. I. O. Arushanyan, “On the numerical solution of boundary integral equations of the second kind in domains with corner points”, Comput. Math. Math. Phys., 36:6 (1996), 773–782
9. E. A. Volkov, A. K. Kornoukhov, E. A. Yakovleva, “Experimental investigation of the block method for the Laplace equation on polygons”, Comput. Math. Math. Phys., 38:9 (1998), 1481–1489
10. E. A. Volkov, A. K. Kornoukhov, “An approximate conformal mapping of a trapezoid onto a rectangle and its inversion obtained by the block method”, Comput. Math. Math. Phys., 39:7 (1999), 1100–1108
11. E. A. Volkov, A. K. Kornoukhov, “Solving the torsion problem for an $L$-section rod by the block method”, Comput. Math. Math. Phys., 42:8 (2002), 1161–1170
12. E. A. Volkov, A. K. Kornoukhov, “On solving the Motz problem by a block method”, Comput. Math. Math. Phys., 43:9 (2003), 1331–1337
13. Zi-Cai Li, Tzon-Tzer Lu, Hsin-Yun Hu, Alexander H.D. Cheng, “Particular solutions of Laplace's equations on polygons and new models involving mild singularities”, Engineering Analysis with Boundary Elements, 29:1 (2005), 59
14. A. A. Dosiyev, S. Cival Buranay, “On solving the cracked-beam problem by block method”, Commun Numer Meth Engng, 24:11 (2007), 1277
15. A.A. Dosiyev, Z. Mazhar, S.C. Buranay, “Block method for problems on L-shaped domains”, Journal of Computational and Applied Mathematics, 235:3 (2010), 805
16. A.A. Dosiyev, S.C. Buranay, D. Subasi, “The highly accurate block-grid method in solving Laplace’s equation for nonanalytic boundary condition with corner singularity”, Computers & Mathematics with Applications, 2012
17. A.A. Dosiyev, Emine Celiker, “Approximation on the hexagonal grid of the Dirichlet problem for Laplace’s equation”, Bound Value Probl, 2014:1 (2014), 73
18. A.A. Dosiyev, S.C. Buranay, “One-block method for computing the generalized stress intensity factors for Laplace’s equation on a square with a slit and on an L-shaped domain”, Journal of Computational and Applied Mathematics, 2014
19. A.A. Dosiyev, Emine Celiker, “A fourth order block-hexagonal grid approximation for the solution of Laplace’s equation with singularities”, Adv Differ Equ, 2015:1 (2015)
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