Numerical Implementation of the Fictitious Domain Method for Elliptic Equations
Abstract
In this paper, we consider an elliptic equation with strongly varying coefficients. Interest in the study of these equations is connected with the fact that this type of equation is obtained when using the fictitious domain method. In this paper, we propose a special method for the numerical solution of elliptic equations with strongly varying coefficients. A theorem is proved for the rate of convergence of the iterative process developed. A computational algorithm and numerical calculations are developed to illustrate the effectiveness of the proposed method.References
A. A. Samarskiy. “About One Economic Algorithm of Numerical Computation of Differential and Algebraic Equation System,” Journal of Computation Mathematics and Mathematical Physics, vol. 4, no. 3, pp. 580-585, 1964 (in Russian).
A. B. Kucherov and E. S. Nikolayev, “Alternately-Triangular Method of Mesh Elliptical Equiations Solution in Rectangle,” Journal of Computation Mathematics and Mathematical Physics, vol. 16, no. 5, pp. 1164-1174, 1976.
V. N. Lebedev, “Method of Composition,” Moscow: OBM. AS USSR, 1986.
E. A. Volkov, ”About Methods Solving Difference Equations for Sectionally-Uniform Media and With the Right Part of Given along Curve”. DAN USSR, vol. 283, no. 2, pp. 274-277, 1985.
A. N. Bugrov, A. N. Konovalov, and V. A. Scherbak, “Fictitious Domain Method in Plane Elastic Static Problems Numerical Methods of Mechanics Of Continua,” Novosibirsk, vol. 5, no. 1, pp.20-30, 1974.
A. N. Konovalov, “Fictitious Domain Method in Torsion Problems Numerical Methods of Mechanics Of Continua,” Novosibirsk, vol. 4, no. 2, pp. 109-115, 1973.
Sh. S. Smagulov, “Fictitious Domain Method for Boundary Problem of Navier-Stokes Equation,” Novosibirsk: Ed. V.Ts. SO AN USSR, no. 68, pp. 68-73, 1979.
M. K. Orunkhanov and Sh. S. Smagulov, “Fictitious Domain Method for Navier-Stokes Equation in Terms of Flow Function and Vorticity with Inhomogeneous Boundary Conditions Calculation Technologies,” Novosibirsk: SO RAN, vol. 5, no.3, pp. 46-53, 2000.
Sh. N. Kuttykozhaeva, “Fictitious Domain Method for Navier-Stokes Equation,” Vestnik KazGU. Section Math., Mech., Inf., no 13, pp. 54-59, 1998.
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