# Preconditioning techniques using algebraic tools

**Preconditioning techniques using algebraic tools**

The purpose of this work is the use of algebraic and symbolic techniques such as Smith normal forms and Grobner basis techniques in order to develop new Schwarz algorithms and preconditionners for linear systems of partial differential equations (PDEs). This work is motivated by the fact that in some sense these methods applied systems of partial differential equations (such as Stokes, Oseen, linear elasticity) are less optimal than the domain decomposition methods for scalar problems. Indeed, in the case of two subdomains consisting of the two half planes it is well known, that the Neumann-Neumann preconditioner is an exact preconditioner for the Schur complement equation for scalar equations like the Laplace problem. A preconditioner is called exact, if the preconditioned operator simplifies to the identity. Unfortunately, this does not hold in the vector case. In order to achieve this goal we use algebraic methods developed in constructive algebra, D-modules (differential modules) and symbolic computation such as the so-called Smith or Jacobson normal forms and Grobner basis techniques for transforming a linear system of PDEs into a set of independent scalar PDEs. Decoupling linear systems of PDEs leads to the design of new numerical methods based on the efficient techniques dedicated to scalar PDEs (e.g., Laplace equation, advection-diffusion equation). Moreover, these algebraic and symbolic methods provide important intrinsic information (e.g., invariants) about the linear system of PDEs to solve which need to be taken into account in the design of new numerical methods which can supersede the usual ones based on a direct extension of the classical scalar methods to the linear systems.