SADDLES

The purpose of the SADDLES project is the use of algebraic and symbolic techniques such as Smith normal forms and Gröbner basis techniques in order to develop new numerical methods for linear systems of partial differential equations (PDEs). The numerical simulation relies heavily on solving systems of PDEs, that can be linear or non linear, time dependent or stationary.

Non-linear problems are solved by fixed point methods or Newton type algorithms, leading themselves to solving linear systems of equations. To solve these systems, the direct methods, based on LU or Cholesky factorization, are often preferred because of their robustness. But for three (or higher) dimension problems, because of their important memory requirements and computational costs, they can be used only when the number of unknowns is not so large. Hence, for the large scale problems we deal with in today's standard applications, it is necessary to rely on iterative Krylov methods, interesting because of their limited memory requirements and scalability properties. They are preconditioned by domain decomposition methods, incomplete factorizations, or multigrid preconditioners. These methods are well understood and efficient for scalar symmetric equations (Laplacian, biLaplacian, ...), and to some extent for non symmetric equations (convection-diffusion, ...). But they exhibit poor performance and lack robustness when they are used for systems of PDEs, especially for the non symmetric case (fluid mechanics, porous media, ...). The goal of the project is to analyze these difficulties and propose solutions. The originality of the proposal lies in the use of a new approach for this analysis, based on algebraic tools like the Smith factorization. Algebraic tools like the Smith factorization enable an intrinsic analysis of systems of partial differential operators. They allow thus the system to be written as a set of decoupled equations. The approach is totally different from the classical factorization of a system of ordinary differential equations. For example, with this approach there is no need to enlarge the system by introducing additional unknowns, and that independently of the orders of differentiation in the system. Moreover the approach is very general and can be applied to very important systems of PDEs such that the Oseen equations (linearized Navier-Stokes equations). The project will use principally the Smith factorization as an algebraic tool, and possibly other algebraic concepts as well. Some specific formal calculus tools need to be developped especially for the discretized equations.