Computational Mechanics using High Performance Computing
Edited by: B.H.V. Topping

Chapter 9

Domain Decomposition Methods for Non-Symmetric Problems

Two algorithms, especially suited to non-symmetric elliptic problems, are presented. The model equation is the convection-diffusion equation. This equation is important in itself in engineering or environnemental sciences for instance, it models the transport and diffusion of species (e.g. pollutant in air or water, electrons in semiconductor devices) in a given flow. It is also a key aspect of the Navier-Stokes equations. An implicit scheme in time will demand at a solution procedure at each time step The first algorithm is a preconditioner for the Schur formulation of domain decomposition problems. It is an extension of the well-known Neumann-Neumann preconditioner to non-symmetric problems. The second algorithm can be seen as a modification of the Schwarz method. The Dirichlet boundary conditions on the interfaces are replaced by more general boundary conditions. The algorithm can then be used on non-overlapping subdomains and has a fast convergence. We emphasize a presentation at the matrix level.

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