Heterogeneity and anisotropy are phenomena encountered in many different settings: flows in porous media, optical tomography, neutron transport, radiative transfer, electromagnetics, etc. While the differential equations modeling these problems may be different in nature, the common feature they share is that they involve parameters that may be highly heterogeneous (values may have jumps of many orders of magnitude) or can be highly anisotropic. Under the usual terminology these problems are degenerate (i.e. there is no uniform ellipticity) and, as a result, cannot be handled using standard mathematical methods. The objective of this research project is to analyze degenerate problems (well-posedness, stability, etc.) and to develop general discontinuous Galerkin techniques for approximating them. The discontinuous Galerkin that will be constructed will be stable for all the degenerate cases of interest and will automatically approximate the physically meaningful transmission conditions between sub-domains with different parameter ranges without the user having to identify sub-domains with specific properties a priori. The key is to use the mathematical framework of Friedrichs systems and to design the operators that controls discontinuities in the approximate solution appropriately.
The merit of the proposed approximation technique is that it addresses the problem at hand from a radically different perspective than the usual multi-domain/multi-algorithmic approaches. The new technique will automatically detect heterogeneity/anisotropy and will adapt to situations without the user having to take manual action. The impact of this research project will be broad since the class of problems addressed touches many fields in engineering, in environmental sciences, in geophysics, in petroleum engineering, semiconductor industry, etc. Proposing a novel robust approximation technique for solving problems with highly heterogeneous and anisotropic properties will eventually benefit many areas of science and engineering where controlling or dealing with this type of problem is still a serious challenge.
