Journal articles on the topic 'Substructuring preconditioner'

AbstractA DD (domain decomposition) preconditioner of almost optimal in p arithmetical complexity is presented for the hierarchical hp discretizations of 3-d second order elliptic equations. We adapt the wire basket substructuring technique to the hierarchical hp discretization, obtain a fast preconditioner-solver for faces by Kinterpolation technique and show that a secondary iterative process may be efficiently used for prolongations from faces. The fast solver for local Dirichlet problems on subdomains of decomposition is based on our earlier derived finite-difference like preconditioner for the internal stiffness matrices of p-finite elements and fast solution procedures for systems with this preconditioner, which appeared recently.

Abstract The discretization of optimal control of elliptic partial differential equations problems yields optimality conditions in the form of large sparse linear systems with block structure. Correspondingly, when the solution method is a Dirichlet-Dirichlet non-overlapping domain decomposition method, we need to solve interface problems which inherit the block structure. It is therefore natural to consider block preconditioners acting on the interface variables for the acceleration of Krylov methods with substructuring preconditioners. In this paper we describe a generic technique which employs a preconditioner block structure based on the fractional Sobolev norms corresponding to the domains of the boundary operators arising in the matrix interface problem, some of which may include a dependence on the control regularization parameter. We illustrate our approach on standard linear elliptic control problems. We present analysis which shows that the resulting iterative method converges independently of the size of the problem. We include numerical results which indicate that performance is also independent of the control regularization parameter and exhibits only a mild dependence on the number of the subdomains.

Summary The major issues for parallel solvers in a modern reservoir simulator are robustness, scalability, efficiency, and flexibility. There is significant interest in running fast field-scale simulations for complex giant Middle Eastern reservoirs, which will require tens of millions to hundreds of millions of grid cells to give reasonable resolution. At the same time, significant geologic complexity will require the treatment of dual-permeability regions, faulting and fractures, and high variations of reservoir and fluid properties. Of course, the methods should also work well for extracted-sector simulation with local grid refinements in both the structured and unstructured discretization. The preconditioning methods considered in this work include both the single-stage and multistage frameworks. In the single-stage framework, a novel method is considered in addition to the well-known variants of incomplete lower-upper (ILU) factorizations [ILU0, ILU(k), and ILUT]. The new method is a highly parallel method, which, in this paper, will be referred to as the unstructured line-solve power-series (LSPS) method. The method will be discussed and contrasted in light of key issues for parallel linear solvers. The unstructured LSPS has certain interesting properties in the parallel construct, which make it a highly effective component. The multistage method researched in this work is of the constraint pressure residual (CPR) framework. The method uses approximate pressure solve as the first-stage preconditioning to the full-system preconditioning. A number of original adaptations based on this concept were researched. Here, the use of the parallel algebraic multigrid (PAMG) method and other single-level methods mentioned previously in combinations within the multistage CPR framework were explored. Certain methods constructed in this way are found to be highly efficient, scalable, and robust. The methods developed are discussed, and several test problems are included, in this paper. The largest simulation model tested to date using these solver methods is a 172-million-cell full-field model of a supergiant carbonate complex with more than 3,000 wells and 60 years of history simulation. Introduction Parallel reservoir simulation involving millions of grid cells is now common practice and is an essential component for the management of many giant carbonate complexes in the Middle East. The recent advances are aided in part by the computational power offered by inexpensive PC clusters. Many of today's parallel machines are built with mass-produced commodity-based components. At the same time, research and development on parallel highly scalable methods in the modern reservoir simulator have made routine field-scale simulation an effective and useful part of resource planning and analysis. Field-scale analyses are often desired over sector simulation for a comprehensive understanding of overall reservoir-behavior and recovery-processes performance. Special study involving an area of interest frequently arises in a full-field project. For example, evaluation of alternative designs for expensive maximum-reservoir-contact wells with intelligent downhole controls and production equipment requires near-wellbore reservoir simulation and optimization workflow. Thus, the demand is high for simulation capabilities with mixed structured and unstructured grids for fast field-scale megacell modeling. The capability to refine and coarsen at ease regionally and perform simulation and analyses at multiple scales within a single project is a primary near-term goal. This paper addresses one critical component of the tool set required to accomplish this mission--the linear solver. The primary solver methods in the old generation of reservoir simulators typically use nested factorization or variants of ILU-factorization method for preconditioning. While extension to small-scale parallel processing was achieved in the late 1990s, these methods have limitations in terms of scalability or robustness for the very-large-scale simulations where parallel processing with hundreds or even thousands of processors is required for speed and performance. Previously, within the structured-grid framework, a solver method known as the z-line Neumann series, which is more scalable for parallel field-scale simulation of structured grid, was documented by Dogru et al. (2002). Later, a parallel structured multigrid method was introduced by Fung and Dogru (2000) for treating the local-grid-refinement problems. The additional solver method for the dual-porosity dual-permeability system was later described by Fung and Al-Shaalan (2005). In this work, new ideas in the fully unstructured setting are being researched and developed. These ideas involve both the single-stage method and the multistage method. In the single-stage method, a novel idea of building an approximate inverse preconditioner through matrix substructuring of the Jacobian matrix was investigated. This substructuring method, which we refer to as LSPS, is a powerful generalization of the z-line Neumann series method. The method is fully unstructured. It increases robustness by tracing the maximum-transmissibility direction of the 3D unstructured graph. The strategy is particularly beneficial for reservoirs with fracture corridors and superpermeability (super-K) regions that cause difficulties for other solver methods. Furthermore, parallel efficiency is maintained, which is crucial for large-scale multiprocessor applications of the method. In the multistage method, the two-stage CPR method was investigated. The CPR method was first introduced into the petroleum literature by Wallis (1983) and Wallis et al. (1985). It was recently applied by Gratien et al. (2004) and Cao et al. (2005) in a new simulator development in which they have used the PAMG method as the pressure preconditioner. The research documented here explores the quasi-implicit-pressure-explicit-saturation (quasi-IMPES) reduction methods and the use of various approaches to solve the pressure approximately as a first-stage preconditioning to the full-system matrix. Solver results for several sample problems are included for comparison of the various methods. These include the public-domain data sets for the SPE1 (Odeh 1981) and SPE10 (Christie and Blunt 2001) comparative-solution projects and several megacell-simulation models. To add some challenge for the solver methods, the SPE1 grid system has been refined uniformly to 300,000 cells. To put all the methods into proper prospective, the three variants of the ILU factorizations [ILU0, ILU(k), and ILUT] are used as baseline comparison for some problems. The ILU preconditioners are well-known and are described in Saad (2003), thus descriptions of them are not included here. Interested readers can refer to Saad (2003) or the many other reference papers concerned with them.