Some algorithmic aspects of systems of PDEs based simulations can be better clarified by means of symbolic computation techniques. Many iterative solvers of linear systems issued from the discretization of (PDEs) use preconditionners such as domain decomposition methods. These techniques are well understood and efficient for scalar symmetric equations  but the study of  iterative solvers for systems of PDEs as opposed to scalar PDEs is an under developed subject.
The aim of this work is to use algebraic techniques to build new robust and efficient algorithms for solving systems of PDEs.
Recents studies have shown how the Smith normal form can help in such problems. In particular, we are interested in transforming the linear system of PDEs into a set of decoupled PDEs under certain types of invertible transformations and then to use efficient existing algorithms for solving scalar equations.
The computation of the Smith normal form S of a matrix R over a univariate polynomial ring, provides two unimodular matrices E and F satisfying R=ESF. For many applications, this issue is not really  important. However, in this talk, we show that the computation of unimodular matrices F whose entries are simple and have physical meanings is an important issue for the reduction of the interface conditions appearing in the domain decomposition algorithms. Then we will show how this allows us to obtain simple reduced interface conditions. Another contribution of the talk is to prove how algebraic techniques (namely normal forms and Grobner bases) can be used to reduce these interface conditions. The algorithms developed have been implemented in Maple using the OreModules and OreMorphisms packages. We will illustrate some examples of their application on the Navier-Cauchy, Stokes and Oseen equations.

This is a work in progress in collaboration with V. Dolean (University of Nice - Sophia-Anitpolis), F. Nataf (University of Paris 6) and A. Quadrat (INRIA Sophia-Antipolis) which is supported by the CNRS via the PEPS SADDLES (http://www-math.unice.fr/~dolean/saddles/)