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