Numerical Solution of Interface and Transmission Problems
by
Boško S. Jovanović
University of Belgrade, Faculty of Mathematics, Studentski trg 16, 11000 Belgrade, Serbia

The transfer of energy and mass is fundamental in many chemical, biological, environmental and industrial processes. The basic transport mechanisms in these processes are diffusion (or dispersion) and bulk flow. Here we focus on diffusion in domains with layers.

Layers with material properties which significantly differ from those of the surrounding medium often appear in a variety of applications. The layer may have a structural role (as in the case of glue), a thermal role (as in the case of thermal insulator), an electromagnetic or optical role, etc.

Problems with thin layers can be modelled by partial differential equations whose input data and the solutions have discontinuities across one or several interfaces (lines, surfaces etc.). Corresponding boundary value problems are commonly called interface problems. Standard numerical methods designed for the problems with smooth solutions do not work efficiently in the case of interface problems.

In the case of thick layers in an analogous way one obtains so called transmission problems, whose solutions are defined in two (or more) disjoint domains. For example, such a situation occurs when the solution in the intermediate region is known, or can be determined from a simpler equation. The effect of the intermediate region can be modelled by means of nonlocal jump conditions.

In this paper some model examples of interface and transmission problems are presented and the corresponding numerical methods for their solution are proposed and investigated.

REFERENCES

1. B.S. Jovanović, L.G. Vulkov, Operator's approach to the problems with concentrated factors, Lect. Notes in Comput. Sci. 1988 (2001), 439-451.

2. B.S. Jovanović, L.G. Vulkov, Finite difference approximations for some interface problems with variable coefficients, Appl. Numer. Math. (2008), to appear.

3. J. Kandilarov, L. Vulkov, The immersed interface method for two-dimensional heat-diffusion equations with singular own sources, Appl. Numer. Math. 57, 5-7 (2007), 486-497.

4. L.G. Vulkov, Well posedness and a monotone iterative method for a nonlinear interface problem on disjoint intervals, Amer. Inst. of Physics, Proceedings Series 946 (2007).

Date received: May 5, 2008