Block monotone domain decomposition methods for a nonlinear anisotropic convection-diffusion equation

Abstract

This article deals with discrete monotone iterative methods for solving a quasi-linear, singularly perturbed, convection-diffusion problem. A block monotone domain decomposition method based on a Schwarz alternating method and on block (line) successive under-relaxation iterative method is constructed. The advantages of this monotone method are that the method solves only linear discrete systems at each iterative step of the iterative process and converges monotonically to the exact solution of the quasi-linear problem. Numerical experiments are presented. References I. Boglaev, A monotone Schwarz algorithm for a semilinear convection-diffusion problem. J. Numer. Math., 12:169--191, 2004. doi:10.1016/j.cam.2004.03.011 I. P. Boglaev, V. V. Sirotkin, A. Ya. Vilenkin, Ch. V. Kopetskii, A. V. Serebryakov, On the numerical simulation of heat transfer in the process of casting amorphous metal ribbons. Preprint, Institute of Solid State Physics, USSR Academy of Sciences, Chernogolovka, 1983. I. Boglaev and S. Pack, On block monotone domain decomposition algorithms for solving a nonlinear singularly perturbed convection-diffusion problem. Reports in Mathematics 17, IFS, Massey University, 2007. T. Chan and T. Mathew, Domain decomposition algorithms. Acta Numerica, 4:61--143, 1994. B. Smith, P. Bjorstad, and W. Gropp, Domain decomposition. Cambridge University Press, Cambridge, 1996. http://www.cambridge.org/0521602866 R. S. Varga, Matrix Iterative Analysis. Second Edition. Springer--Verlag, Berlin Heidelberg, 2000.