The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Summary. In previous works [21–23] we proposed the use of [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned conjugate gradient (PCG) methods. As and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and, in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations [36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence. On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth. Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis. Received December 19, 1995 / Revised version received September 15, 1997     

Local refinement techniques for elliptic problems on cell-centered grids; II. Optimal order two-grid iterative methods

Two preconditioning techniques for solving difference equations arising in finite difference approximation of elliptic problems on cell-centered grids are studied. It is proven that the BEPS and the FAC preconditioners are spectrally equivalent to the corresponding finite difference schemes, including a nonsymmetric one, which is of higher-order accuracy. Numerical experiments that demonstrate the fast convergence of the preconditioned iterative methods (CG and GCG-LS in the nonsymmetric case) are presented.     

Superlinearly convergent PCG algorithms for some nonsymmetric elliptic systems

A preconditioned conjugate gradient method is applied to finite element discretizations of some nonsymmetric elliptic systems. Mesh independent superlinear convergence is proved, which is an extension of a similar earlier result from a single equation to systems. The proposed preconditioning method involves decoupled preconditioners, which yields small and parallelizable auxiliary problems.     

Experimental study of ILU preconditioners for indefinite matrices

Incomplete LU factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices. However, their failure rate is still too high for them to be useful as black-box library software for general matrices. Besides fatal breakdowns due to zero pivots, the major causes of failure are inaccuracy, and instability of the triangular solves. When there are small pivots, both these problems can occur, but these problems can also occur without small pivots. Through examples from actual problems, this paper shows how these problems evince themselves, how these problems can be detected, and how these problems can sometimes be circumvented through pivoting, reordering, scaling, perturbing diagonal elements, and preserving symmetric structure. The goal of this paper is to gain a better practical understanding of ILU preconditioners and help improve their reliability.     

A class of generalized shift-splitting preconditioners for nonsymmetric saddle point problems

In this paper, a class of generalized shift-splitting preconditioners with two shift parameters are implemented for nonsymmetric saddle point problems with nonsymmetric positive definite (1, 1) block. The generalized shift-splitting (GSS) preconditioner is induced by a generalized shift-splitting of the nonsymmetric saddle point matrix, resulting in an unconditional convergent fixed-point iteration. By removing the shift parameter in the (1, 1) block of the GSS preconditioner, a deteriorated shift-splitting (DSS) preconditioner is presented. Some useful properties of the DSS preconditioned saddle point matrix are studied. Finally, numerical experiments of a model Navier–Stokes problem are presented to show the effectiveness of the proposed preconditioners.     

Overlapping Schwarz preconditioners for spectral element methods in nonstandard domains and heterogeneous media

Overlapping Schwarz preconditioners are constructed and numerically studied for Gauss-Lobatto-Legendre (GLL) spectral element discretizations of heterogeneous elliptic problems on nonstandard domains defined by Gordon-Hall transfinite mappings. The results of several test problems in the plane show that the proposed preconditioners retain the good convergence properties of overlapping Schwarz preconditioners for standard affine GLL spectral elements, i.e. their convergence rate is independent of the number of subdomains, of the spectral degree in the case of generous overlap and of the discontinuity jumps in the coefficients of the elliptic operator, while in the case of small overlap, the convergence rate depends on the inverse of the overlap size.     

A class of constraint preconditioners for nonsymmetric saddle point matrices

We consider the use of a class of constraint preconditioners for the application of the Krylov subspace iterative method to the solution of large nonsymmetric, indefinite linear systems. The eigensolution distribution of the preconditioned matrix is determined and the convergence behavior of a Krylov subspace method such as GMRES is described. The choices of the parameter matrices and the implementation of the preconditioning step are discussed. Numerical experiments are presented. This work is supported by NSFC Projects 10171021 and 10471027.     

Superlinear PCG Algorithms: Symmetric Part Preconditioning and Boundary Conditions

The superlinear convergence of the preconditioned CGM is studied for nonsymmetric elliptic problems (convection-diffusion equations) with mixed boundary conditions. A mesh independent rate of superlinear convergence is given when symmetric part preconditioning is applied to the FEM discretizations of the BVP. This is the extension of a similar result of the author for Dirichlet problems. The discussion relies on suitably developed Hilbert space theory for linear operators.     

On block preconditioners for saddle point problems with singular or indefinite (1, 1) block

We discuss a class of preconditioning methods for the iterative solution of symmetric algebraic saddle point problems, where the (1, 1) block matrix may be indefinite or singular. Such problems may arise, e.g. from discrete approximations of certain partial differential equations, such as the Maxwell time harmonic equations. We prove that, under mild assumptions on the underlying problem, a class of block preconditioners (including block diagonal, triangular and symmetric indefinite preconditioners) can be chosen in a way which guarantees that the convergence rate of the preconditioned conjugate residuals method is independent of the discretization mesh parameter. We provide examples of such preconditioners that do not require additional scaling. Copyright © 2010 John Wiley & Sons, Ltd.     

Strang‐type preconditioners applied to ordinary and neutral differential‐algebraic equations

This paper deals with boundary‐value methods (BVMs) for ordinary and neutral differential‐algebraic equations. Different from what has been done in Lei and Jin (Lecture Notes in Computer Science, vol. 1988. Springer: Berlin, 2001; 505–512), here, we directly use BVMs to discretize the equations. The discretization will lead to a nonsymmetric large‐sparse linear system, which can be solved by the GMRES method. In order to accelerate the convergence rate of GMRES method, two Strang‐type block‐circulant preconditioners are suggested: one is for ordinary differential‐algebraic equations (ODAEs), and the other is for neutral differential‐algebraic equations (NDAEs). Under some suitable conditions, it is shown that the preconditioners are invertible, the spectra of the preconditioned systems are clustered, and the solution of iteration converges very rapidly. The numerical experiments further illustrate the effectiveness of the methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Preconditioning techniques for a mixed Stokes/Darcy model in porous media applications

We study numerical methods for a mixed Stokes/Darcy model in porous media applications. The global model is composed of two different submodels in a fluid region and a porous media region, coupled through a set of interface conditions. The weak formulation of the coupled model is of a saddle point type. The mixed finite element discretization applied to the saddle point problem leads to a coupled, indefinite, and nonsymmetric linear system of algebraic equations. We apply the preconditioned GMRES method to solve the discrete system and are particularly interested in efficient and effective decoupled preconditioning techniques. Several decoupled preconditioners are proposed. Theoretical analysis and numerical experiments show the effectiveness and efficiency of the preconditioners. Effects of physical parameters on the convergence performance are also investigated.     

Eigenvalues of the discretized Navier-Stokes equation with application to the detection of Hopf bifurcations

This paper is concerned with a numerical approach to the problem of finding the leftmost eigenvalues of large sparse nonsymmetric generalised eigenvalue problems which arise in stability studies of incompressible fluid flow problems. The matrices have a special block structure that is typical of mixed finite element discretizations for such problems. The numerical approach is an extension of the hybrid technique introduced by Saad [22] and utilizes the idea of preconditioning the eigenvalue problem before applying Arnoldi's method. Two preconditioners, one a modified Cayley transform, the other a Chebyshev polynomial transform, are compared in numerical experiments on a double diffusive convection problem and the Cayley transform proves superior. The Cayley transform is then used to provide numerical results for the finite Taylor problem.     

On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems

In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, we present block SSOR and modified block SSOR iteration methods based on the special structures of the coefficient matrices. In each step of the block SSOR iteration, we employ the block LU factorization to solve the sub-systems of linear equations. We show that the block LU factorization is existent and stable when the coefficient matrices are block diagonally dominant of type-II by columns. Under suitable conditions, we establish convergence theorems for both block SSOR and modified block SSOR iteration methods. In addition, the block SSOR iteration and AF-ADI method are considered as preconditioners for the nonsymmetric systems of linear equations. Numerical experiments show that both block SSOR and modified block SSOR iterations are feasible iterative solvers and they are also effective for preconditioning Krylov subspace methods such as GMRES and BiCGSTAB when used to solve this class of systems of linear equations.     

On the performance of the algebraic optimized Schwarz methods with applications

We investigate the performance of algebraic optimized Schwarz methods used as preconditioners for the solution of discretized differential equations. These methods consist on modifying the so-called transmission blocks. The transmission blocks are replaced by new blocks in order to improve the convergence of the corresponding iterative algorithms. In the optimal case, convergence in two iterations can be achieved. We are also interested in the behavior of the algebraic optimized Schwarz methods with respect to changes in the problems parameters. We focus on constructing preconditioners for different numerically challenging differential problems such as: Periodic and Torus problems; Meshfree problems; Three-dimensional problems. We present different numerical simulations corresponding to different type of problems in two- and three-dimensions.     

Block ILU Preconditioners for a Nonsymmetric Block-Tridiagonal M-Matrix

We propose block ILU (incomplete LU) factorization preconditioners for a nonsymmetric block-tridiagonal M-matrix whose computation can be done in parallel based on matrix blocks. Some theoretical properties for these block ILU factorization preconditioners are studied and then we describe how to construct them effectively for a special type of matrix. We also discuss a parallelization of the preconditioner solver step used in nonstationary iterative methods with the block ILU preconditioners. Numerical results of the right preconditioned BiCGSTAB method using the block ILU preconditioners are compared with those of the right preconditioned BiCGSTAB using a standard ILU factorization preconditioner to see how effective the block ILU preconditioners are.     

A comparison of preconditioned Krylov subspace methods for large‐scale nonsymmetric linear systems

Preconditioned Krylov subspace (KSP) methods are widely used for solving large‐scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While implementations of preconditioned KSP methods are usually readily available, it is unclear to users which methods are the best for different classes of problems. In this work, we present a comparison of some KSP methods, including GMRES, TFQMR, BiCGSTAB, and QMRCGSTAB, coupled with three classes of preconditioners, namely, Gauss–Seidel, incomplete LU factorization (including ILUT, ILUTP, and multilevel ILU), and algebraic multigrid (including BoomerAMG and ML). Theoretically, we compare the mathematical formulations and operation counts of these methods. Empirically, we compare the convergence and serial performance for a range of benchmark problems from numerical PDEs in two and three dimensions with up to millions of unknowns and also assess the asymptotic complexity of the methods as the number of unknowns increases. Our results show that GMRES tends to deliver better performance when coupled with an effective multigrid preconditioner, but it is less competitive with an ineffective preconditioner due to restarts. BoomerAMG with a proper choice of coarsening and interpolation techniques typically converges faster than ML, but both may fail for ill‐conditioned or saddle‐point problems, whereas multilevel ILU tends to succeed. We also show that right preconditioning is more desirable. This study helps establish some practical guidelines for choosing preconditioned KSP methods and motivates the development of more effective preconditioners.     

Algebraic multilevel iteration method for Stieltjes matrices

The numerical solution of elliptic selfadjoint second-order boundary value problems leads to a class of linear systems of equations with symmetric, positive definite, large and sparse matrices which can be solved iteratively using a preconditioned version of some algorithm. Such differential equations originate from various applications such as heat conducting and electromagnetics. Systems of equations of similar type can also arise in the finite element analysis of structures. We discuss a recursive method constructing preconditioners to a symmetric, positive definite matrix. An algebraic multilevel technique based on partitioning of the matrix in two by two matrix block form, approximating some of these by other matrices with more simple sparsity structure and using the corresponding Schur complement as a matrix on the lower level, is considered. The quality of the preconditioners is improved by special matrix polynomials which recursively connect the preconditioners on every two adjoining levels. Upper and lower bounds for the degree of the polynomials are derived as conditions for a computational complexity of optimal order for each level and for an optimal rate of convergence, respectively. The method is an extended and more accurate algebraic formulation of a method for nine-point and mixed five- and nine-point difference matrices, presented in some previous papers.     

A comparison of some domain decomposition and ILU preconditioned iterative methods for nonsymmetric elliptic problems

In recent years, competitive domain-decomposed preconditioned iterative techniques of Krylov-Schwarz type have been developed for nonsymmetric linear elliptic systems. Such systems arise when convection-diffusion-reaction problems from computational fluid dynamics or heat and mass transfer are linearized for iterative solution. Through domain decomposition, a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow effective solution on parallel machines. A central question is how to choose these small problems and how to arrange the order of their solution. Different specifications of decomposition and solution order lead to a plethora of algorithms possessing complementary advantages and disadvantages. In this report we compare several methods, including the additive Schwarz algorithm, the classical multiplicative Schwarz algorithm, an accelerated multiplicative Schwarz algorithm, the tile algorithm, the CGK algorithm, the CSPD algorithm, and also the popular global ILU-family of preconditioners, on some nonsymmetric or indefinite elliptic model problems discretized by finite difference methods. The preconditioned problems are solved by the unrestarted GMRES method. A version of the accelerated multiplicative Schwarz method is a consistently good performer.