Constructive Approximations of the Convection-Diffusion-Absoption Equation Based on the Cayley Transform Technique

Gavriljuk, I.P. and Melnik, R.V.N.

Computational Mechanics. New Trends and Applications. Proceedings of the Forth World Congress on Computational Mechanics, Eds. S.R. Idelsohn, E. Onate, E.N. Dvorkin, ISBN 84-89925-15-1, 14 pages, 1998

Abstract:

In many practical problems the norm of the resolvent for the convection-diffusion -absorption (CDA) operator is large as a function of the Peclet number. As a result, conventional spectral analysis may be of limited usefulness for the solution of such problems. Fluid dynamics and electromagnetic theory, along with other areas of applications, provide a wide range of this type of problems. Even if the spatial operator is spectrally discretised, the solution of nonstationary problems requires a temporal discretisation that dictates a typically severe restriction on the method applicability in convection-dominated regions. To resolve these difficulties we develop a new method based on a combination of the Cayley transform technique and an effective iterated mapping. Error estimates for our numerical method are presented.

Key words: CDA operators in Banach spaces; Cayley transform; iterated mapping; spectral angle; Cauchy-Riesz integrals; error estmates; non-selfadjointness; time-dependent problems; resolvent bounds.