Partial Differential Equations with Tensor Coefficients
C. Farmer (Schlumberger Abingdon Technology Center & University of Oxford)
Partial differential equations with tensor coefficients, where the tensor is not diagonal in the coordinate system of the problem, arise in many applications. In particular they are important in problems of flow through porous media. Even equations with isotropic coefficients can be problematic in a similar way, when one is forced to use non-orthogonal grids.
The recent development of the multi-point flux and related finite volume methods has enabled the problems to be solved in many cases, but difficulties still occur when the off-diagonal terms are large. Practical application of these methods requires improvements in robustness and understanding, so that the full benefits of tensors can be exploited. Reductions in simulation grid size will be possible if anisotropic media can be modelled at the large scale, after homogenisation, rather than requiring fine scale simulation with equivalent isotropic properties. Improved models of the geometry of systems will be possible if those involved in grid generation are set free to generate non-orthogonal or other extreme grids.
This minisymposium provides an opportunity to review progress and hear about some new ideas. Improvements in this subject are of enormous importance in the optimal management of hydrocarbon and water resources. Thus this minisymposium will provide another example of the value of mathematical analysis for industrial applications.