Preconditioning of weighted H(div)-norm and applications to numerical simulation of highly heterogeneous media

In this paper we propose and analyze a preconditioner for a system arising from a finite element approximation of second order elliptic problems describing processes in highly het- erogeneous media. Our approach uses the technique of multilevel methods and the recently proposed preconditioner based on additive Schur complement approximation by J. Kraus (see [8]). The main results are the design and a theoretical and numerical justification of an iterative method for such problems that is robust with respect to the contrast of the media, defined as the ratio between the maximum and minimum values of the coefficient (related to the permeability/conductivity).Comment: 28 page

A class of multilevel parallel preconditioning strategies

In this paper, we introduce a class of recursive multilevel preconditioning strategies suited for solving large sparse linear systems of equations on modern day architectures. They are based on a reordering of the input matrix into a nested bordered block diagonal form, which allows a nested formulation of the preconditioners. The first one, which we refer to as nested SSOR (NSSOR), requires only the factorization of diagonal blocks at the innermost level of the recursive formulation. Hence, its construction is embarassingly parallel, and the memory requirements are very limited. Next two are nested versions of Modified ILU preconditioner with row sum (NMILUR) and colsum (NMILUC) property. We compare these methods in terms of iteration number, memory requirements, and overall solve time, with ILU(0) with natural ordering and nested dissection ordering, and MILU. We find that NSSOR compares favorably with ILU(0) with nested dissection ordering, while NMILUR and NMILUC outperform the other methods for certain matrices in our test set. It is proved that the NSSOR method is convergent when the input matrix is SPD. The preconditioners are designed to be suitable for parallel computing.Dans ce papier nous décrivons une classe de préconditionneurs multiniveaux parallèles pour résoudre des systèmes linéaires de grande taille. Ils se basent sur une renumérotation de la matrice d'entrée en forme block diagonale bornée et emboitée, qui permet une définition emboitée des préconditionneurs. Nous prouvons qu'un des préconditionneurs, NSSOR, converge quand la matrice d'entrée est symmétrique et définie positive. Les préconditionneurs sont adaptés au calcul parallèle

Multilevel Variable-Block Schur-Complement-Based Preconditioning for the Implicit Solution of the Reynolds- Averaged Navier-Stokes Equations Using Unstructured Grids

Implicit methods based on the Newton’s rootfinding algorithm are receiving an increasing attention for the solution of complex Computational Fluid Dynamics (CFD) applications due to their potential to converge in a very small number of iterations. This approach requires fast convergence acceleration techniques in order to compete with other conventional solvers, such as those based on artificial dissipation or upwind schemes, in terms of CPU time. In this chapter, we describe a multilevel variable-block Schur-complement-based preconditioning for the implicit solution of the Reynolds-averaged Navier-Stokes equations using unstructured grids on distributed-memory parallel computers. The proposed solver detects automatically exact or approximate dense structures in the linear system arising from the discretization, and exploits this information to enhance the robustness and improve the scalability of the block factorization. A complete study of the numerical and parallel performance of the solver is presented for the analysis of turbulent Navier-Stokes equations on a suite of three-dimensional test cases

Preconditioning of block tridiagonal matrices

Preconditioning methods via approximate block factorization for block tridiagonal matrices are studied. Bounds for the resulting condition numbers are given, and two methods for the recursive construction of the approximate Schur complements are presented. Illustrations for elliptic problems are also given, including a study of sensitivity to jumps in the coefficients and of a suitably motidied Poincaré-Steklov operator on the continuous level