Numerical Experiments With An Overlapping Additive Schwarz Solver For 3-D Parallel Reservoir Simulation

. Domain decomposition methods are a major area of contemporary research in numerical analysis of partial differential equations. They provide robust, parallel and scalable preconditioned iterative methods for the large linear systems arising when continuous problems are discretized by finite elements, finite differences or spectral methods. This paper presents some numerical experiments on a distributed memory parallel computer, the 512 processor Caltech Touchstone Delta. An overlapping additive Schwarz method is implemented for the mixed finite element discretization of second order elliptic problems in three dimensions, arising from flow models in reservoir simulation. These problems are characterized by large variations in the coefficients of the elliptic operator, often associated with short correlation lengths, which make the problems very ill-conditioned. The results confirm the theoretical bound on the condition number of the iteration operator and show the advantage of domain ..

Scalability of preconditioners as a strategy for parallel computation of compressible fluid flow

Parallel implementations of a Newton-Krylov-Schwarz algorithm are used to solve a model problem representing low Mach number compressible fluid flow over a backward-facing step. The Mach number is specifically selected to result in a numerically {open_quote}stiff{close_quotes} matrix problem, based on an implicit finite volume discretization of the compressible 2D Navier-Stokes/energy equations using primitive variables. Newton`s method is used to linearize the discrete system, and a preconditioned Krylov projection technique is used to solve the resulting linear system. Domain decomposition enables the development of a global preconditioner via the parallel construction of contributions derived from subdomains. Formation of the global preconditioner is based upon additive and multiplicative Schwarz algorithms, with and without subdomain overlap. The degree of parallelism of this technique is further enhanced with the use of a matrix-free approximation for the Jacobian used in the Krylov technique (in this case, GMRES(k)). Of paramount interest to this study is the implementation and optimization of these techniques on parallel shared-memory hardware, namely the Cray C90 and SGI Challenge architectures. These architectures were chosen as representative and commonly available to researchers interested in the solution of problems of this type. The Newton-Krylov-Schwarz solution technique is increasingly being investigated for computational fluid dynamics (CFD) applications due to the advantages of full coupling of all variables and equations, rapid non-linear convergence, and moderate memory requirements. A parallel version of this method that scales effectively on the above architectures would be extremely attractive to practitioners, resulting in efficient, cost-effective, parallel solutions exhibiting the benefits of the solution technique