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The highly accurate block-grid method in solving Laplace’s equation for nonanalytic boundary condition with corner singularity. (English) Zbl 1252.65204
Summary: The highly accurate block-grid method for solving Laplace’s boundary value problems on polygons is developed for nonanalytic boundary conditions of the first kind. The quadrature approximation of the integral representations of the exact solution around each reentrant corner(“singular” part) are combined with the 9-point finite difference equations on the “nonsingular” part. In the integral representations, and in the construction of the sixth order gluing operator, the boundary conditions are taken into account with the help of integrals of Poisson type for a half-plane which are computed with $ϵ$ accuracy. It is proved that the uniform error of the approximate solution is of order $O\left({h}^{6}+ϵ\right)$, where $h$ is the mesh step. This estimation is true for the coefficients of singular terms also. The errors of $p$-order derivatives ($p=0,1,\cdots$) in the “singular” parts are $O\left(\left({h}^{6}+ϵ\right){r}_{j}^{1/{\alpha }_{j}-p}\right)$, ${r}_{j}$ is the distance from the current point to the vertex in question and ${\alpha }_{j}\pi$ is the value of the interior angle of the $j$th vertex. Finally, we give the numerical justifications of the obtained theoretical results.
##### MSC:
 65N99 Numerical methods for BVP of PDE
##### References:
 [1] Andreev, V. B.: Asymptotic solution of Laplace’s grid equation in a corner, Dokl. akad. Nauk SSSR 244, 1289-1293 (1979) [2] Brenner, S. C.: Multigrid methods for the computation of singular solutions and stress intensity factors I: Corner singularities, Math. comp. 68, 559-583 (1999) · Zbl 1043.65136 · doi:doi:10.1090/S0025-5718-99-01017-0 [3] Li, Z. C.; Lu, T. T.: Singularities and treatments of elliptic boundary problem, Math. comput. Modelling 31, 97-145 (2000) · Zbl 1042.35524 · doi:doi:10.1016/S0895-7177(00)00062-5 [4] Dosiyev, A. A.: The high accurate block-grid method for solving Laplace’s boundary value problem with singularities, SIAM J. Numer. anal. 42, No. 1, 153-178 (2004) · Zbl 1080.65103 · doi:doi:10.1137/S0036142900382715 [5] Xenophontos, C.; Elliotis, M.; Georgiou, G.: A singular function boundary integral method for Laplacian problems with boundary singularities, SIAM J. Sci. comput. 28, No. 2, 517-532 (2006) · Zbl 1120.65121 · doi:doi:10.1137/050622742 [6] Chein-Shan, Liu: A highly accurate solver for the mixed-boundary potential problem and singular problem in arbitrary plane domain, CMES comput. Model. eng. Sci. 20, No. 2, 111-122 (2007) · Zbl 1184.65107 · doi:http://techscience.com/cmes/2007/v20n2_index.html [7] Li, Z. C.: Combined methods for elliptic problems with singularities, interfaces and infinities, (1998) [8] Volkov, E. A.; Dosiyev, A. A.: A high accurate composite grid method for solving Laplace’s boundary value problems with singularities, Russian J. Numer. anal. Math. modelling 22, No. 3, 291-307 (2007) · Zbl 1135.65040 · doi:doi:10.1515/RJNAMM.2007.014 [9] Fix, G.: Higher order Rayleigh–Ritz approximations, J. math. Mech. 18, 645-658 (1969) [10] 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, No. 2, 396-399 (1992) · Zbl 0795.65066 [11] 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, No. 5, 591-604 (1994) · Zbl 0832.65113 [12] Dosiyev, A. A.; Buranay, S. C.: A fourth order accurate difference-analytical method for solving Laplace’s boundary value problem with singularities, Mathematical methods in engineers, 167-176 (2007) · Zbl 1130.65105 · doi:doi:10.1007/978-1-4020-5678-9_13 [13] Dosiyev, A. A.; Buranay, S. Cival; Subasi, D.: The block-grid method for solving Laplace’s equation on polygons with nonanalytic boundary conditions, Bound. value probl. 2010 (2010) · Zbl 1214.65054 · doi:doi:10.1155/2010/468594 [14] Wu, X.; Han, H.: A finite-element method for Laplace-and Helmholtz-type boundary value problems with singularities, SIAM J. Numer. anal. 34, No. 3, 1037-1050 (1997) · Zbl 0873.65100 · doi:doi:10.1137/S0036142993258488 [15] Dosiyev, A. A.: On the maximum error in the solution of Laplace equation by finite difference method, Int. J. Pure appl. Math. 7, No. 2, 229-241 (2003) · Zbl 1057.65075 [16] Volkov, E. A.: Approximate solution of Laplace’s equation by the block method on polygons under nonanalytical boundary conditions, Proc. Steklov inst. Math., No. 4, 65-90 (1993) · Zbl 0801.65103 [17] Volkov, E. A.: Block method for solving the Laplace equation and constructing conformal mappings, (1994) · Zbl 0914.65112 [18] Brown, J. V.; Churchill, R. V.: Complex variables and applications, (1996) [19] Volkov, E. A.: An exponentially converging method for solving Laplace’s equation on polygons, Math. USSR sb. 37, No. 3, 295-325 (1980) · Zbl 0444.35031 · doi:doi:10.1070/SM1980v037n03ABEH001954 [20] Volkov, E. A.: On differentiability properties of solutions of boundary value problems for the Laplace’s equation on polygons, Tr. mat. Inst. akad. Nauk SSSR 77, 113-142 (1965) [21] Volkov, E. A.: Development of the block method for solving the Laplace equation for finite and infinite circular polygons, Trudy mat. Inst. Steklov 187, 39-68 (1989) · Zbl 0708.65096 [22] Kondratiev, V. A.: Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow math. Soc. 16, 227-313 (1967) · Zbl 0194.13405 [23] Arad, M.; Yosibash, Z.; Ben-Dor, G.; Yakhot, A.: Computing flux intensity factors by a boundary method for elliptic equations with singularities, Comm. numer. Methods engrg. 14, 657-670 (1998) · Zbl 0911.65103 · doi:doi:10.1002/(SICI)1099-0887(199807)14:7<657::AID-CNM180>3.0.CO;2-K [24] Kelley, C. T.; Sachs, E. W.: Mesh independence of Newton-like methods for infinite dimensional problems, J. integral equations appl. 3, No. 4, 549-573 (1991) · Zbl 0756.65085 · doi:doi:10.1216/jiea/1181075649