Volume 11, no. 2Pages 123 - 138

Development and Analysis of the Fast Pseudo Spectral Method for Solving Nonlinear Dirichlet Problems

B.V. Semisalov
Numerical method for solving one-, two- and three-dimensional Dirichlet problems for the nonlinear elliptic equations has been designed. The method is based on the application of Chebyshev approximations without saturation and on a new way of forming and solving the systems of linear equations after discretization of the original differential problem. Wherein the differential operators are approximated by means of matrices and the equation itself is approximated by the Sylvester equation (2D case) or by its tensor generalization (3D case). While solving test problems with the solutions of different regularity we have shown a rigid correspondence between the rate of convergence of the proposed method and the order of smoothness (or regularity) of the sought-for function. The observed rates of convergence strictly correspond to the error estimates of the best polynomial approximations and show the absence of saturation of the designed algorithm. This results in the essential reduction of memory costs and number of operations for cases of the problems with solutions of a high order of smoothness.
Full text
Chebyshev approximation; boundary-value problem; nonlocal method without saturation; stabilization method.
1. Babenko K.I. Osnovy chislennogo analiza [Fundamentals of Numerical Analysis]. Moscow, Nauka, 1986. 714 p.
2. Babenko K.I. [On the Saturation Phenomenon in Numerical Analysis]. Dokl. Akad. Nauk SSSR [Doklady Mathematics], 1978, vol. 241, no. 3, pp. 505-508. (in Russian)
3. Semisalov B.V. Non-Local Algorithm of Finding Solution to the Poisson Equation and Its Applications. Computational Mathematics and Mathematical Physics, 2014, vol. 54, no. 7, pp. 1110-1135. (in Russian)
4. Trefethen L.N. Approximation Theory and Approximation Practice. Philadelphia, SIAM, 2013.
5. Dzjadyk V.K. Vvedenie v teoriju ravnomernogo priblizhenija funkcij polinomami [Introduction to the Theory of Uniform Approximation by Polynomials]. Moscow, Nauka, 1977.
6. Blokhin A.M., Ibragimova A.S. Numerical Method for 2D Simulation of a Silicon MESFET with a Hydrodynamical Model Based on the Maximum Entropy Principle. SIAM Journal Scientific Computing, 2009, vol. 31, no. 3, pp. 2015-2046. DOI: 10.1137/070706537
7. Belov A.A., Kalitkin N.N. Evolutionary Factorization and Superfast Relaxation Count. Mathematical Models and Computer Simulations, 2014, vol. 26, no. 9, pp. 47-64. (in Russian)
8. Belykh V.N. Particular Features of Implementation of an Unsaturated Numerical Method for the Exterior Axisymmetric Neumann Problem. Siberian Mathematical Journal, 2013, vol. 54, no. 6, pp. 984-993. DOI: 10.1134/S0037446613060037
9. Boyd J. Chebyshev and Fourier Spectral Methods. Mineola, N.Y., University of Michigan, 2000.