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Analysis of the block-grid method for the solution of Laplace’s equation on polygons with a slit. (English) Zbl 1470.65184

Summary: The error estimates obtained for solving Laplace’s boundary value problem on polygons by the block-grid method contain constants that are difficult to calculate accurately. Therefore, the experimental analysis of the method could be essential. The real characteristics of the block-grid method for solving Laplace’s equation on polygons with a slit are analysed by experimental investigations. The numerical results obtained show that the order of convergence of the approximate solution is the same as in the case of a smooth solution. To illustrate the singular behaviour around the singular point, the shape of the highly accurate approximate solution and the figures of its partial derivatives up to second order are given in the “singular” part of the domain. Finally a highly accurate formula is given to calculate the stress intensity factor, which is an important quantity in fracture mechanics.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N22 Numerical solution of discretized equations for boundary value problems involving PDEs
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[1] Li, Z. C., Combined Methods for Elliptic Problems with Singularities, Interfaces and Infinities, (1998), London, UK: Kluwer Academic, London, UK · Zbl 0909.65079
[2] Dosiyev, A. A., A block-grid method for increasing accuracy in the solution of the Laplace equation on polygons, Russian Academy of Sciences, 45, 2, 396-399, (1992)
[3] Dosiyev, A. A., A block-grid method of increased accuracy for solving Dirichlet’s problem for Laplace’s equation on polygons, Computational Mathematics and Mathematical Physics, 34, 5, 591-604, (1994) · Zbl 0832.65113
[4] Dosiyev, A. A., The high accurate block-grid method for solving Laplace’s boundary value problem with singularities, SIAM Journal on Numerical Analysis, 42, 1, 153-178, (2004) · Zbl 1080.65103
[5] Dosiyev, A. A.; Cival, S., A difference-analytical method for solving Laplace’s boundary value problem with singularities, Proceedings of Conference on Applications of Dynamical Systems
[6] Dosiyev, A. A.; Buranay, S. C.; Tas, K.; Machado, J. A. T.; Baleanu, D., A fourth order accurate difference-analytical method for solving Laplace’s boundary value problem with singularities, Mathematical Methods in Engineers, 167-176, (2007), Berlin, Germany: Springer, Berlin, Germany · Zbl 1130.65105
[7] Volkov, E. A., An exponentially converging method for solving Laplace’s equation on polygons, Mathematics of the USSR-Sbornik, 37, 3, 295-325, (1980) · Zbl 0444.35031
[8] Volkov, E. A., Block Method for Solving the Laplace Equation and Constructing Conformal Mappings, (1994), Boca Raton, Fla, USA: CRC Press, Boca Raton, Fla, USA · Zbl 0914.65112
[9] Wigley, N. M., An efficient method for subtracting off singularities at corners for Laplace’s equation, Journal of Computational Physics, 78, 2, 369-377, (1988) · Zbl 0657.65129
[10] Fix, G. J.; Gulati, S.; Wakoff, G. I., On the use of singular functions with finite element approximations, Journal of Computational Physics, 13, 209-228, (1973) · Zbl 0273.35004
[11] Wasow, W., On the truncation error in the solution of Laplace’s equation by finite differences, Journal of Research of the National Bureau of Standards, 48, 345-348, (1952)
[12] Romanova, S. E., An efficient method for the approximate solution of the Laplace difference equation on rectangular domains, Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 23, 3, 660-673, (1983) · Zbl 0526.65073
[13] Dosiyev, A. A.; Buranay, S. C., On solving the cracked-beam problem by block method, Communications in Numerical Methods in Engineering with Biomedical Applications, 24, 11, 1277-1289, (2008) · Zbl 1153.74039
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