A class of nonsymmetric preconditioners for saddle point problems

For iterative solution of saddle point problems, a nonsymmetric preconditioning is studied which, with respect to the upper-left block of the system matrix, can be seen as a variant of SSOR. An idealized situation where the SSOR is taken with respect to the skew-symmetric part plus the diagonal part of the upper-left block is analyzed in detail. Since action of the preconditioner involves solution of a Schur complement system, an inexact form of the preconditioner can be of interest. This results in an inner-outer iterative process. Numerical experiments with solution of linearized Navier-Stokes equations demonstrate efficiency of the new preconditioner, especially when the left-upper block is far from symmetric

Preconditioners for Generalized Saddle-Point Problems

Generalized saddle point problems arise in a number of applications, ranging from optimization and metal deformation to fluid flow and PDE-governed optimal control. We focus our discussion on the most general case, making no assumption of symmetry or definiteness in the matrix or its blocks. As these problems are often large and sparse, preconditioners play a critical role in speeding the convergence of Krylov methods for these problems. We first examine two types of preconditioners for these problems, one block-diagonal and one indefinite, and present analyses of the eigenvalue distributions of the preconditioned matrices. We also investigate the use of approximations for the Schur complement matrix in these preconditioners and develop eigenvalue analysis accordingly. Second, we examine new developments in probing methods, inspired by graph coloring methods for sparse Jacobians, for building approximations to Schur complement matrices. We then present an analysis of these techniques and their accuracy. In addition, we provide a mathematical justification for their use in approximating Schur complements and suggest the use of approximate factorization techniques to decrease the computational cost of applying the inverse of the probed matrix. Finally, we consider the effect of our preconditioners on four applications. Two of these applications come from the realm of fluid flow, one using a finite element discretization and the other using a spectral discretization. The third application involves the stress relaxation of aluminum strips at low stress levels. The final application involves mesh parameterization and flattening. For these applications, we present results illustrating the eigenvalue bounds on our preconditioners and demonstrating the theoretical justification of these methods. We also present convergence and timing results, showing the effectiveness of our methods in practice. Specifically the use of probing methods for approximating the Schur compliment matrices in our preconditioners is empirically justified. We also investigate the $h$-dependence of our preconditioners one model fluid problem, and demonstrate empirically that our methods do not suffer from a deterioration in convergence as the problem size increases

Efficient linear solvers for incompressible flow simulations using Scott--Vogelius finite elements

Recent research has shown that in some practically relevant situations like multiphysics flows (Galvin et al., Comput Methods Appl Mech Eng, 2012) divergence-free mixed finite elements may have a significantly smaller discretization error than standard nondivergence-free mixed finite elements. To judge the overall performance of divergence-free mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in Scott-Vogelius finite element implementations of the incompressible Navier-Stokes equations. We investigate both direct and iterative solver methods. Due to discontinuous pressure elements in the case of Scott-Vogelius (SV) elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements such as Taylor-Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes and discuss extensions. For iterative methods, we test augmented Lagrangian and H -LU preconditioners with GMRES, on both full and statically condensed systems. Several numerical experiments are provided that show these classes of solvers are well suited for use with SV elements and could deliver an interesting overall performance in several applications

A linear multigrid preconditioner for the solution of the Navier-Stokes equations using a discontinuous Galerkin discretization

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 69-72).A Newton-Krylov method is developed for the solution of the steady compressible Navier-Stokes equations using a Discontinuous Galerkin (DG) discretization on unstructured meshes. An element Line-Jacobi preconditioner is presented which solves a block tridiagonal system along lines of maximum coupling in the flow. An incomplete block-LU factorization (Block-ILU(O)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling used for the element Line-Jacobi preconditioner. This reordering is shown to be far superior to standard reordering techniques (Nested Dissection, One-way Dissection, Quotient Minimum Degree, Reverse Cuthill-Mckee) especially for viscous test cases. The Block-ILU(0) factorization is performed in-place and a novel algorithm is presented for the application of the linearization which reduces both the memory and CPU time over the traditional dual matrix storage format. A linear p-multigrid algorithm using element Line-Jacobi, and Block-ILU(O) smoothing is presented as a preconditioner to GMRES.(cont.) The coarse level Jacobians are obtained using a simple Galerkin projection which is shown to closely approximate the linearization of the restricted problem except for perturbations due to artificial dissipation terms introduced for shock capturing. The linear multigrid preconditioner is shown to significantly improve convergence in terms of the number of linear iterations as well as to reduce the total CPU time required to obtain a converged solution. A parallel implementation of the linear multi-grid preconditioner is presented and a grid repartitioning strategy is developed to ensure scalable parallel performance.by Laslo Tibor Diosady.S.M

Refined isogeometric analysis: a solver-based discretization method

112 p.Isogeometric Analysis (IGA) is a computational approach frequently employed nowadaysto study problems governed by partial differential equations (PDEs). This approach definesthe geometry using conventional CAD functions and, in particular, NURBS. Thesefunctions represent complex geometries commonly found in engineering design and arecapable of preserving exactly the geometry description under refinement as required in theanalysis. Moreover, the use of NURBS as basis functions is compatible with theisoparametric concept, allowing to build algebraic systems directly from the computationaldomain representation based on spline functions, which arise from CAD. Therefore, itavoids to define a second space for the numerical analysis resulting in huge reductions inthe total analysis time.For the case of direct solvers, the performance strongly depends upon the employeddiscretization method. In particular, on IGA, the continuity of the solution spaces plays asignificant role in their performance. High continuous spaces degrade the direct solver'sperformance, increasing the solution times by a factor up to O(p^3) with respect totraditional finite element analysis (FEA) per unknown, being p the polynomial order.In this work, we propose a solver-based discretization that employs highly continuous finiteelement spaces interconnected with low continuity hyperplanes to maximize theperformance of direct solvers. Starting from a highly continuous IGA discretization, weintroduce C^0 hyperplanes, which act as separators for the direct solver, to reduce theinterconnection between the degrees of freedom (DoF) in the mesh. By doing so, both thesolution time and best approximation errors are simultaneously improved. We call theresulting method ``refined Isogeometric analysis" (rIGA). Numerical results indicate thatrIGA delivers speed-up factors proportional to p^2. For instance, in a 2D mesh with fourmillion elements and p=5, a Laplace linear system resulting from rIGA is solved 22 timesfaster than the one from highly continuous IGA. In a 3D mesh with one million elementsand p=3, the linear rIGA system is solved 15 times faster than the IGA one.We have also designed and implemented a similar rIGA strategy for iterative solvers. Thisis a hybrid solver strategy that combines a direct solver (static condensation step) toeliminate the internal macro-elements DoF, with an iterative method to solve the skeletonsystem. The hybrid solver strategy achieves moderate savings with respect to IGA whensolving a 2D Poisson problem with a structured mesh and a uniform polynomial degree ofapproximation. For instance, for a mesh with four million elements and polynomial degreep=3, the iterative solver is approximately 2.6 times faster (in time) when applied to the rIGAsystem than to the IGA one. These savings occur because the skeleton rIGA systemcontains fewer non-zero entries than the IGA one. The opposite situation occurs for 3Dproblems, and as a result, 3D rIGA discretizations provide no gains with respect to theirIGA counterparts.Thesis director(s): David Pardo from UPV/EHU university and Victor M. Calo from Curtinuniversit