Symmetric Positive Systems Research Articles

A novel direct resolution method for coupled systems in finite element analysis

PurposeThis paper aims to propose a novel direct method for indefinite algebraic linear systems. It is well adapted for sparse linear systems, such as those of two-dimensional (2-D) finite elements problems, especially for coupled systems.Design/methodology/approachThe proposed method is developed on an example of an indefinite symmetric matrix. The algorithm of the method is given next, and a comparison between the numbers of operations required by the method and the Cholesky method is also given. Finally, an application on a magnetostatic problem for classical methods (Gauss and Cholesky) shows the relative efficiency of the proposed method.FindingsThe proposed method can be used advantageously for 2-D finite elements in stepping methods without using a block decomposition of matrices.Research limitations/implicationsThis method is advantageous for direct linear solving for 2-D problems, but it is not recommended at this time for three-dimensional problems.Originality/valueThe proposed method is the first direct solver for algebraic linear systems proposed since more than a half century. It is not limited for symmetric positive systems such as many of direct and iterative methods.

Cached Gaussian elimination for simulating Stokes flow on domains with repetitive geometry

Microfluidic “lab on a chip” devices are small devices that operate on small length scales on small volumes of fluid; these devices find uses in a variety of applications. Designs for microfluidic chips are generally composed of standardized and often repeated components connected by long, thin, straight fluid channels. We propose a novel meshing algorithm for use in simulating the linear incompressible stationary Stokes equations on geometry with these features, which produces sparse symmetric positive indefinite systems with many repeated matrix blocks. We use a discretization that is formally third order accurate for velocity and second order accurate for pressure in the L∞ norm. We also propose a novel linear system solver based on cyclic reduction, reordered sparse Gaussian elimination, and operation caching that is designed to efficiently solve systems with repeated matrix blocks. We demonstrate that the resulting fluid solver is significantly faster than existing methods up to resolutions of a few million degrees of freedom for microfluidic problems.

The Well-Posedness of the Cauchy Problem for the Dirac Operator on Globally Hyperbolic Manifolds with Timelike Boundary

We consider the Dirac operator on globally hyperbolic manifolds with timelike boundary and show well-posedness of the Cauchy initial-boundary value problem coupled to MIT-boundary conditions. This is achieved by transforming the problem locally into a symmetric positive hyperbolic system, proving existence and uniqueness of weak solutions and then using local methods developed by Lax, Phillips and Rauch, Massey to show smoothness of the solutions. Our proof actually works for a slightly more general class of local boundary conditions.

Solving the Stokes Problem on a Massively Parallel Computer

Abstract We describe a numerical procedure for solving the stationary two‐dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distri...

Avoiding singular coarse grid systems

Here we consider the iterative solution of linear systems of equations with a symmetric positive semidefinite system matrix. If multilevel methods in combination with Krylov subspace methods are used to find the solution of these systems, often singular subsystems or coarse grid systems have to be solved. Then the Moore–Penrose inverse of the coarse grid systems can be used. Here, we establish some theoretical techniques how to avoid the singularity of the coarse grid system, while the resulting operator remains the same as we would have used the Moore–Penrose inverse. One option is to delete specific columns of the restriction and prolongation operator. The other option is to perturb the system matrix.

Non-Stationary Abstract Friedrichs Systems

Inspired by the results of Ern et al. (Commun Partial Differ Equ 32:317–341, 2007) on the abstract theory for Friedrichs symmetric positive systems, we give the existence and uniqueness result for the initial- (boundary) value problem for the non-stationary abstract Friedrichs system. Despite the absence of the well-posedness result for such systems, there were already attempts for their numerical treatment by Burman et al. (SIAM J Numer Anal 48:2019–2042, 2010) and Bui-Thanh et al. (SIAM J Numer Anal 51:1933–1958, 2013). We use the semigroup theory approach and prove that the operator involved satisfies the conditions of the Hille–Yosida generation theorem. We also address the semilinear problem and apply the new results to a number of examples, such as the symmetric hyperbolic system, the unsteady div–grad problem, and the wave equation. Special attention was paid to the (generalised) unsteady Maxwell system.

Deflation and projection methods applied to symmetric positive semi-definite systems

Linear systems with a singular symmetric positive semi-definite matrix appear frequently in practice. This usually does not lead to difficulties for CG methods as long as these systems are consistent. However, the construction of a preconditioner, especially the construction of two-level and multilevel methods, becomes more complicated, since singular coarse grid matrices or Galerkin matrices may occur. Here we continue the work started in [21,22] where deflation is used for some special singular coefficient matrices. Here we show that deflation and other projection-type preconditioners can be applied to arbitrary singular problems without any difficulties. In each of these methods, a two-level preconditioner is involved where coarse-grid systems based on a singular Galerkin matrix should be solved. We prove that each projection operator consisting of a singular Galerkin matrix can be written as an operator with a nonsingular Galerkin matrix. Therefore many results that hold for nonsingular Galerkin matrices are also valid for problems with singular Galerkin matrices.

An Iterative Method for Symmetric Positive Semidefinite Linear System of Equations

Abstract In this paper, a new two-step iterative method for solving symmetric positive semidefinite linear system of equations is presented. A sufficient condition for the semiconvergence of the method is also given. Some numerical experiments are presented to show the efficiency of the proposed method.

Heat equation as a Friedrichs system

Inspired by recent advances in the theory of (Friedrichs) symmetric positive systems, we apply newly developed results to the heat equation, by showing how the intrinsic theory of Ern, Guermond and Caplain (2007) can be used in order to get a well-posedness result for the Dirichlet initial–boundary value problem. We also demonstrate the application of the two-field theory with partial coercivity of Ern and Guermond (2008), originally developed for elliptic problems, and also discuss different possibilities for the construction of the appropriate boundary operator.

Second-order equations as Friedrichs systems

On the basis of the recent progress in understanding the abstract setting for Friedrichs symmetric positive systems by Ern et al. (2007) [8], as well as Antonić and Burazin (2010) [9], an attempt is made to relate these results to the classical Friedrichs theory.A particular set of sufficient conditions for a boundary matrix field to define a boundary operator is given, and the applicability of this procedure is shown by examples of boundary/initial value problems for second-order partial differential equations written as symmetric systems.

Connecting Classical and Abstract Theory of Friedrichs Systems via Trace Operator

Based on recent progress in understanding the abstract setting for Friedrichs symmetric positive systems by Ern et al. (2007), as well as Antonić and Burazin (2010), we continue our efforts to relate these results to the classical Friedrichs theory. Following the approach via the trace operator, we extend the results of Antonić and Burazin (2011) to situations where the important boundary field does not consist only of projections, allowing the treatment of hyperbolic equations, besides the elliptic ones.

Semiconvergence of parallel multisplitting methods for symmetric positive semidefinite linear systems

In this paper we construct some parallel relaxed multisplitting methods for solving consistent symmetric positive semidefinite linear systems, based on modified diagonally compensated reduction and incomplete factorizations. The semiconvergence of the parallel multisplitting method, relaxed multisplitting method and relaxed two-stage multisplitting method are discussed. The results generalize some well-known results for the nonsingular linear systems to the singular systems. Copyright © 2010 John Wiley & Sons, Ltd.

Boundary operator from matrix field formulation of boundary conditions for Friedrichs systems

Following the recent progress in understanding the abstract setting for Friedrichs symmetric positive systems by Ern, Guermond and Caplain (2007) [8], as well as Antonić and Burazin (2010) [3], an attempt is made to relate these results to the classical Friedrichs theory. A comparison of two approaches, via the trace operator and the boundary operator, has been made, favouring the latter. Finally, a particular set of sufficient conditions for a boundary matrix field to define a boundary operator in that case is given, and the applicability of this procedure in realistic situations is shown by examples.

Intrinsic Boundary Conditions for Friedrichs Systems

The admissible boundary conditions for symmetric positive systems of first-order linear partial differential equations, originally introduced by Friedrichs [11], were recently related to three different sets of intrinsic geometric conditions in graph spaces [10]. We rewrite their cone formalism in terms of an indefinite inner product space, which in a quotient by its isotropic part gives a Kreǐn space. This new viewpoint allows us to show that the three sets of intrinsic boundary conditions are actually equivalent, which will hopefully facilitate further investigation of their precise relation to the original Friedrichs boundary conditions.

An Intrinsic Criterion for the Bijectivity of Hilbert Operators Related to Friedrich' Systems

Friedrich' theory of symmetric positive systems of first-order PDE's is revisited so as to avoid invoking traces at the boundary. Two intrinsic geometric conditions are introduced to characterize admissible boundary conditions. It is shown that the space in which admissible boundary conditions can be enforced is maximal in a positive cone associated with the differential operator. The equivalence with a formalism based on boundary operators is investigated and practical means to construct these boundary operators are presented. Finally, the link with Friedrich' formalism and applications to various PDE's are discussed.

Erratum: “The helically-reduced wave equation as a symmetric-positive system” [J. Math. Phys. 44, 6623 (2003)

First Page

Partial differential-algebraic systems of second order with symmetric convection

This paper deals with initial boundary value problems (IBVPs) of linear and some semilinear partial differential algebraic equations (PDAEs) with symmetric first order (convection) terms which are semidiscretized with respect to the space variables by means of a standard conform finite element method. The aim is to give L 2 -convergence results for the semidiscretized systems when the finite element mesh parameter h goes to zero. In general, without the assumption of symmetry (and some further conditions) it is difficult to get such results. According to many practical applications, the PDAEs may have also hyperbolic parts. These are described by means of Friedrichs' theory for symmetric positive systems of differential equations. The PDAEs are assumed to be of time index 1.

The helically-reduced wave equation as a symmetric-positive system

Motivated by the partial differential equations of mixed type that arise in the reduction of the Einstein equations by a helical Killing vector field, we consider a boundary value problem for the helically-reduced wave equation with an arbitrary source in 2+1 dimensional Minkowski space–time. The reduced equation is a second-order partial differential equation which is elliptic in a disk and hyperbolic outside the disk. We show that the reduced equation can be cast into symmetric-positive form. Using results from the theory of symmetric-positive differential equations, we show that this form of the helically-reduced wave equation admits unique, strong solutions for a class of boundary conditions which include Sommerfeld conditions at the outer boundary.

SOLVING THE STOKES PROBLEM ON A MASSIVELY PARALLEL COMPUTER

We describe a numerical procedure for solving the stationary two-dimensional Stokes problem based on piecewise linear finite element approximations for both velocity and pressure, a regularization technique for stability, and a defect‐correction technique for improving accuracy. Eliminating the velocity unknowns from the algebraic system yields a symmetric positive semidefinite system for pressure which is solved by an inner‐outer iteration. The outer iterations consist of the unpreconditioned conjugate gradient method. The inner iterations, each of which corresponds to solving an elliptic boundary value problem for each velocity component, are solved by the conjugate gradient method with a preconditioning based on the algebraic multi‐level iteration (AMLI) technique. The velocity is found from the computed pressure. The method is optimal in the sense that the computational work is proportional to the number of unknowns. Further, it is designed to exploit a massively parallel computer with distributed memory architecture. Numerical experiments on a Cray T3E computer illustrate the parallel performance of the method.

Convergence of two-stage iterative methods for singular symmetric positive semidefinite systems*

Convergence of two-stage iterative methods for singular symmetric positive semidefinite (spsd) systems is studied. The main tool we used to derive the iterative methods and to analyze their convergence is the extended diagonally compensated reduction