Colloquium: Mikolaj Szydlarski, Institute of Theoretical Astrophysics (UiO): Algebraic Domain Decomposition for Darcy flow in heterogeneous media
Abstract: When problems of multiphase, compositional porous media flow simulations are discreetized they lead to complex non linear systems with ill-conditioned matrixes. Their solution by an iterative Krylov method such as GMRES requires the construction of an efficient preconditioner which should be scalable with respect to the heterogeneities, anisotropies of the media, the mesh size and the number of processors. Domain Decomposition Methods (DDM) are an alternative solution for such problems. However when the coefficients of a problem have jumps of several orders of magnitude and are anisotropic such methods suffer from plateaux in the convergence due to the presence of very small isolated eigenvalues in the spectrum of the preconditioned linear system. Optimized Schwarz Method (OSM) is an alternative application of Domain decomposition methods which could solve above difficulties. This method achieve very good performance by using new transmission condition between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem. In my presentation I demonstrate a way to introduce OSM at the matrix level with an inflated system, and I suggest several algebraic approximations of optimal interface conditions to accelerate iterative process. The analysis of various numerical experiments with different Interface conditions ends up with the conclusion that method combined with Improved Interface Condition (IIC) seems to be the best choice in this case.
Coffee/Tea/Biscuits from 14.00.