Volume 22, Issue 4

In this paper we consider a mixed finite element discretization for the diffusion equation on special nonmatching distorted hexahedral meshes. The model problem is motivated by applications in geosciences. We assume that the computational domain is presented as a union of two subdomains separated by, generally speaking, a nonplanar surface, the so-called fault interface. Each subdomain is divided into several 'horizontal' layers corresponding to the media with different material properties. We construct logically cubic hexahedral meshes in each subdomain, which are conforming on the interfaces between different layers but are not obliged to match on the fault interface. We perform a special fault interface reconstruction algorithm in order to construct a conforming polyhedral mesh in the whole domain. Finally, we discretize the problem by the mixed finite element method invented in [2]. The numerical results demonstrate good accuracy of the proposed method.

SparSol is a software package intended for preconditioned iterative solution of large sparse linear systems. It includes a number of iterative methods, preconditioners, scaling and reordering algorithms allowing one to choose the optimal combination of algorithms for a particular problem. The paper briefly describes the implemented algorithms and numerically compares the performance of SparSol with that of several popular, freely available software packages using test systems arising from hydrocarbon reservoir simulations.

In this paper we consider several techniques for construction of general-purpose ILU preconditionings. We also provide some new theoretical results supporting the proposed preconditioning strategy. For the preprocession stage we consider row/column norm-equalizing two-side scaling, as well as diagonal dominance improvement by unsymmetric permutations. Within the incomplete LU-factorization, we use a variant of the inverse norm-reducing Complete Diagonal Pivoting. Numerical results are given for 67 test problems chosen from publicly available matrix sets.

We develop and analyze a new two-level preconditioner for algebraic systems arising from finite volume discretization of 3D anisotropic diffusion problems on prismatic meshes. The preconditioner is based on partitioning of the mesh in the (x,y)-plane into non-overlapping subdomains and on a special coarsening algorithm. The condition number of the preconditioned matrix does not depend on the coefficients in the diffusion operator. Numerical experiments confirm the theoretical results and reveal the competitiveness of the new preconditioner with respect to the algebraic multigrid preconditioner.

This paper describes a family of Nested Factorization (NF) algorithms, which are used as preconditioners in the iterative solution of sparse linear systems arising in discretization of elliptic or parabolic PDE in 2D and 3D. Nested factorization differs from the widely used incomplete LU/Cholesky factorizations in that the preconditioning matrix in NF is not formed strictly from upper and lower factors. Instead, NF exploits the nested triangular structure of the original matrix by defining the preconditioner matrix as a sequence of level matrices with the diagonal matrix on the lowest level. We present the variants of NF and Generalized NF (GNF) for systems arising in regular 7-point stencil and block 7-point stencil discretizations of PDE on structured grids, respectively. We also introduce a variant of Nested Factorization named UNF, applied to matrices arising in discretization of PDE on unstructured grid. This variant is based on a special reordering technique which recovers a multi-level structure for a general sparse matrix. Numerical results presented in the paper show that nested factorization algorithms give faster convergence and require less memory than ILU-type methods for sets of test matrices.

Preconditioned iterative methods provide some of the most efficient approaches for solving general, large sparse linear systems arising in many scientific and engineering applications. It is well-known that the convergence rate of an iterative solver depends strongly on the quality of the preconditioner used. Incomplete LU factorization is one of the most popular preconditioning techniques for unsymmetric linear systems. We present a Fill-in Incomplete LU (FILU) preconditioner which includes two drop tolerance parameters, a fill-in threshold parameter, a strip tolerance parameter, a relaxation parameter, and a set of row norms. The first drop tolerance parameter is used for dropping only new entries, which extends the original matrix pattern. The second drop tolerance parameter is used for dropping entries from a preconditioned row. The fill-in threshold parameter is used to limit the number of nonzero entries in a preconditioned row. The strip tolerance parameter is used for dropping 'small' entries from the original row before preconditioning. The relaxation parameter is a coefficient for subtracting a sum of all dropped entries from the diagonal entry of the corresponding row. And finally, we have developed special row norms different for the L and U parts of a preconditioned row. This extended set of dropping criteria with a special row norm allows us to obtain a preconditioner with a good balance between the quality of LU factorization and the extension ratio of the original pattern. In combination with scalings and reorderings this preconditioner shows good results for problems arising in hydrocarbon reservoir simulation. The performance of the preconditioner is illustrated by numerical experiments on a set of test linear systems and compared with that of several freely available software packages.