Symmetric Saddle Point Systems Research Articles

Deflation for the Off-Diagonal Block in Symmetric Saddle Point Systems

Deflation for the Off-Diagonal Block in Symmetric Saddle Point Systems

A null-space approach for large-scale symmetric saddle point systems with a small and non zero (2, 2) block

Null-space methods have long been used to solve large sparse n × n symmetric saddle point systems of equations in which the (2, 2) block is zero. This paper focuses on the case where the (1, 1) block is ill conditioned or rank deficient and the k × k (2, 2) block is non zero and small (k ≪ n). Additionally, the (2, 1) block may be rank deficient. Such systems arise in a range of practical applications. A novel null-space approach is proposed that transforms the system matrix into a nicer symmetric saddle point matrix of order n that has a non zero (2, 2) block of order at most 2k and, importantly, the (1, 1) block is symmetric positive definite. Success of any null-space approach depends on constructing a suitable null-space basis. We propose methods for wide matrices having far fewer rows than columns with the aim of balancing stability of the transformed saddle point matrix with preserving sparsity in the (1, 1) block. Linear least squares problems that contain a small number of dense rows are an important motivation and are used to illustrate our ideas and to explore their potential for solving large-scale systems.

Convergence of Some Iterative Methods for Symmetric Saddle Point Linear Systems

We consider the iterative solution of linear systems with a symmetric saddle point system matrix. We address a family of solution techniques that exploit the knowledge of a preconditioner (or appro...

A comparative study of null‐space factorizations for sparse symmetric saddle point systems

SummaryNull‐space methods for solving saddle point systems of equations have long been used to transform an indefinite system into a symmetric positive definite one of smaller dimension. A number of independent works in the literature have identified that we can interpret a null‐space method as a matrix factorization. We review these findings, highlight links between them, and bring them into a unified framework. We also investigate the suitability of using null‐space factorizations to derive sparse direct methods and present numerical results for both practical and academic problems.

Sharp stability and approximation estimates for symmetric saddle point systems

We establish sharp well posedness and approximation estimates for variational saddle point systems at the continuous level. The main results of this note have been known to be true only in the finite dimensional case. Known spectral results from the discrete case are reformulated and proved using a functional analysis view, making the proofs in both cases, discrete and continuous, less technical than the known discrete approaches. We focus on analyzing the special case when the form is bounded, symmetric, and coercive, and the mixed form is bounded and satisfies a standard or LBB condition. We characterize the spectrum of the symmetric operators that describe the problem at the continuous level. For a particular choice of the inner product on the product space of , we prove that the spectrum of the operator representing the system at continuous level is . As consequences of the spectral description, we find the minimal length interval that contains the ratio between the norm of the data and the norm of the solution, and prove explicit approximation estimates that depend only on the continuity constant and the continuous and the discrete condition constants

Cascadic multilevel algorithms for symmetric saddle point systems

In this paper, we introduce a multilevel algorithm for approximating variational formulations of symmetric saddle point systems. The algorithm is based on availability of families of stable finite element pairs and on the availability of fast and accurate solvers for symmetric positive definite systems. On each fixed level an efficient solver such as the gradient or the conjugate gradient algorithm for inverting a Schur complement is implemented. The level change criterion follows the cascade principle and requires that the iteration error be close to the expected discretization error. We prove new estimates that relate the iteration error and the residual for the constraint equation. The new estimates are the key ingredients in imposing an efficient level change criterion. The first iteration on each new level uses information about the best approximation of the discrete solution from the previous level. The theoretical results and experiments show that the algorithms achieve optimal or close to optimal approximation rates by performing a non-increasing number of iterations on each level. Even though numerical results supporting the efficiency of the algorithms are presented for the Stokes system, the algorithms can be applied to a large class of boundary value problems, including first order systems that can be reformulated at the continuous level as symmetric saddle point problems, such as the Maxwell equations.

Multilevel Gradient Uzawa Algorithms for Symmetric Saddle Point Problems

In this paper, we introduce a general multilevel gradient Uzawa algorithm for symmetric saddle point systems. We compare its performance with the performance of the standard Uzawa multilevel algorithm. The main idea of the approach is to combine a double inexact Uzawa algorithm at the continuous level with a gradient type algorithm at the discrete level. The algorithm is based on the existence of a priori multilevel sequences of nested approximation pairs of spaces, but the family does not have to be stable. To ensure convergence, the process has to maintain an accurate representation of the residuals at each step of the inexact Uzawa algorithm at the continuous level. The residual representations at each step are approximated by projections or representation operators. Sufficient conditions for ending the iteration on a current pair of discrete spaces are determined by computing simple indicators that involve consecutive iterations. When compared with the standard Uzawa multilevel algorithm, our proposed algorithm has the advantages of automatically selecting the relaxation parameter, lowering the number of iterations on each level, and improving on running time. By carefully choosing the discrete spaces and the projection operators, the error for the second component of the solution can be significantly improved even when comparison is made with the discretization on standard families of stable pairs.

Multilevel discretization of symmetric saddle point systems without the discrete LBB condition

Using an inexact Uzawa algorithm at the continuous level, we study the convergence of multilevel algorithms for solving saddle-point problems. The discrete stability Ladyshenskaya–Babušca–Brezzi (LBB) condition does not have to be satisfied. The algorithms are based on the existence of a multilevel sequence of nested approximation spaces for the constrained variable. The main idea is to maintain an accurate representation of the residual associated with the main equation at each step of the inexact Uzawa algorithm at the continuous level. The residual representation is approximated by a Galerkin projection. Whenever a sufficient condition for the accuracy of the representation fails to be satisfied, the representation of the residual is projected on the next (larger) space available in the prescribed multilevel sequence. Numerical results supporting the efficiency of the algorithms are presented for the Stokes equations and a div–curl system.

FETI and BDD preconditioners for Stokes–Mortar–Darcy Systems

We consider the coupling across an interface of a fluid flow and a porous media flow. The differential equations involve Stokes equations in the fluid region, Darcy equations in the porous region, plus a coupling through an interface with Beaver‐Joseph‐Saffman transmission conditions. The discretization consists of P2/P1 triangular Taylor‐Hood finite elements in the fluid region, the lowest order triangular Raviart‐Thomas finite elements in the porous region, and the mortar piecewise constant Lagrange multipliers on the interface. We allow for nonmatching meshes across the interface. Due to the small values of the permeability parameter of the porous medium, the resulting discrete symmetric saddle point system is very ill conditioned. We design and analyze preconditioners based on the finite element by tearing and interconnecting (FETI) and balancing domain decomposition (BDD) methods and derive a condition number estimate of order C1.1C.1=// for the preconditioned operator. In case the fluid discretization is finer than the porous side discretization, we derive a better estimate of order C2.. C 1/=. C.h p / 2 // for the FETI preconditioner. Here h p is the mesh size of the porous side triangulation. The constants C1 and C2 are independent of the permeability , the fluid viscosity , and the mesh ratio across the interface. Numerical experiments confirm the sharpness of the theoretical estimates.