Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices

For a class of block two-by-two systems of linear equations with certain skew-Hamiltonian coefficient matrices, we construct additive block diagonal preconditioning matrices and discuss the eigen-properties of the corresponding preconditioned matrices. The additive block diagonal preconditioners can be employed to accelerate the convergence rates of Krylov subspace iteration methods such as MINRES and GMRES. Numerical experiments show that MINRES preconditioned by the exact and the inexact additive block diagonal preconditioners are effective, robust and scalable solvers for the block two-by-two linear systems arising from the Galerkin finite-element discretizations of a class of distributed control problems.     

Rotated block triangular preconditioning based on PMHSS

Based on the PMHSS preconditioning matrix, we construct a class of rotated block triangular preconditioners for block two-by-two matrices of real square blocks, and analyze the eigen-properties of the corresponding preconditioned matrices. Numerical experiments show that these rotated block triangular preconditioners can be competitive to and even more efficient than the PMHSS pre-conditioner when they are used to accelerate Krylov subspace iteration methods for solving block two-by-two linear systems with coefficient matrices possibly of nonsymmetric sub-blocks.     

Finite-element based sparse approximate inverses for block-factorized preconditioners

In this work we analyse a method to construct numerically efficient and computationally cheap sparse approximations of some of the matrix blocks arising in the block-factorized preconditioners for matrices with a two-by-two block structure. The matrices arise from finite element discretizations of partial differential equations. We consider scalar elliptic problems, however the approach is appropriate also for other types of problems such as parabolic problems or systems of equations. The technique is applicable for both selfadjoint and non-selfadjoint problems, in two as well as in three space dimensions. We analyse in detail the two-dimensional case and provide extensive numerical evidence for the efficiency of the proposed matrix approximations, both serial and parallel. Two- and three-dimensional tests are included.     

三类有效预处理子的关系及其优化

针对一类由时谐抛物方程约束的最优控制问题导出的分块$2\times2$复线性方程组,进一步研究了三类有效的块预处理子,推导了这三类预处理子间的关系,结论表明三个预处理矩阵的特征值由同一个矩阵确定.通过分析预处理矩阵的谱性质,获得了有效的参数选择策略,可以进一步改进和优化现有结果,同时获得了预处理矩阵的精确特征值分布,并证明了此结果是目前文献中最优结果.最后,给出实例,不仅验证了优化的预处理子和迭代方法的有效性,而且说明了理论结果是令人信服的.     

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 Preconditioning of Incompressible Non-Newtonian Flow Problems

This paper deals with fast and reliable numerical solution methods for the incompressible non-Newtonian Navier-Stokes equations. To handle the nonlinearity of the governing equations, the Picard and Newton methods are used to linearize these coupled partial differential equations. For space discretization we use the finite element method and utilize the two-by-two block structure of the matrices in the arising algebraic systems of equations. The Krylov subspace iterative methods are chosen to solve the linearized discrete systems and the development of computationally and numerically efficient preconditioners for the two-by-two block matrices is the main concern in this paper. In non-Newtonian flows, the viscosity is not constant and its variation is an important factor that affects the performance of some already known preconditioning techniques. In this paper we examine the performance of several preconditioners for variable viscosity applications, and improve them further to be robust with respect to variations in viscosity.     

BCCB preconditioners for systems of BVM‐based numerical integrators

Boundary value methods (BVMs) for ordinary differential equations require the solution of non‐symmetric, large and sparse linear systems. In this paper, these systems are solved by using the generalized minimal residual (GMRES) method. A block‐circulant preconditioner with circulant blocks (BCCB preconditioner) is proposed to speed up the convergence rate of the GMRES method. The BCCB preconditioner is shown to be invertible when the BVM is A_k1,k2‐stable. The spectrum of the preconditioned matrix is clustered and therefore, the preconditioned GMRES method converges fast. Moreover, the operation cost in each iteration of the preconditioned GMRES method by using our BCCB preconditioner is less than that required by using block‐circulant preconditioners proposed earlier. In numerical experiments, we compare the number of iterations of various preconditioners. Copyright © 2003 John Wiley & Sons, Ltd.     

On SSOR‐like preconditioners for non‐Hermitian positive definite matrices

We construct, analyze, and implement SSOR‐like preconditioners for non‐Hermitian positive definite system of linear equations when its coefficient matrix possesses either a dominant Hermitian part or a dominant skew‐Hermitian part. We derive tight bounds for eigenvalues of the preconditioned matrices and obtain convergence rates of the corresponding SSOR‐like iteration methods as well as the corresponding preconditioned GMRES iteration methods. Numerical implementations show that Krylov subspace iteration methods such as GMRES, when accelerated by the SSOR‐like preconditioners, are efficient solvers for these classes of non‐Hermitian positive definite linear systems. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust AMLI methods for parabolic Crouzeix-Raviart FEM systems

For the iterative solution of linear systems of equations arising from finite element discretization of elliptic problems there exist well-established techniques to construct numerically efficient and computationally optimal preconditioners. Among those, most often preferred choices are Multigrid methods (geometric or algebraic), Algebraic MultiLevel Iteration (AMLI) methods, Domain Decomposition techniques.In this work, the method in focus is AMLI. We extend its construction and the underlying theory over to systems arising from discretizations of parabolic problems, using non-conforming finite element methods (FEM). The AMLI method is based on an approximated block two-by-two factorization of the original system matrix. A key ingredient for the efficiency of the AMLI preconditioners is the quality of the utilized block two-by-two splitting, quantified by the so-called Cauchy-Bunyakowski-Schwarz (CBS) constant, which measures the abstract angle between the two subspaces, associated with the two-by-two block splitting of the matrix.The particular choice of space discretization for the parabolic equations, used in this paper, is Crouzeix-Raviart non-conforming elements on triangular meshes. We describe a suitable splitting of the so-arising matrices and derive estimates for the associated CBS constant. The estimates are uniform with respect to discretization parameters in space and time as well as with respect to coefficient and mesh anisotropy, thus providing robustness of the method.