ON THE BREAKDOWNS OF THE GALERKIN AND LEAST-SQUARES METHODS

1　ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｃｏｎｓｉｄｅｒｌｉｎｅａｒｓｙｓｔｅｍｓｏｆｔｈｅｆｏｒｍＡｘ=ｂ,(1 )ｗｈｅｒｅＡ∈ＣＮ×Ｎｉｓｎｏｎｓｉｎｇｕｌａｒａｎｄｐｏｓｓｉｂｌｙｎｏｎ Ｈｅｒｍｉｔｉａｎ .Ａｍａｊｏｒｃｌａｓｓｏｆｍｅｔｈｏｄｓｆｏｒｓｏｌｖｉｎｇ (1 )ｉｓｔｈｅｃｌａｓｓｏｆＫｒｙｌｏｖｓｕｂｓｐａｃｅｍｅｔｈｏｄｓ (ｓｅｅ[6] ,[1 3]ｆｏｒｏｖｅｒｖｉｅｗｓｏｆｓｕｃｈｍｅｔｈｏｄｓ) ,ｄｅｆｉｎｅｄｂｙｔｈｅｐｒｏｐｅｒｔｉｅｓｘｍ ∈ｘ0 +Ｋｍ(ｒ0 ,Ａ) ;(2 )ｒｍ ⊥Ｌｍ, (3)ｗｈｅ…     

New convergence results on the global GMRES method for diagonalizable matrices

In the present paper, we give some new convergence results of the global GMRES method for multiple linear systems. In the case where the coefficient matrix A is diagonalizable, we derive new upper bounds for the Frobenius norm of the residual. We also consider the case of normal matrices and we propose new expressions for the norm of the residual.     

Convergence of CG and GMRES on a tridiagonal Toeplitz linear system

The Conjugate Gradient method (CG), the Minimal Residual method (MINRES), or more generally, the Generalized Minimal Residual method (GMRES) are widely used to solve a linear system Ax=b. The choice of a method depends on A’s symmetry property and/or definiteness), and MINRES is really just a special case of GMRES. This paper establishes error bounds on and sometimes exact expressions for residuals of CG, MINRES, and GMRES on solving a tridiagonal Toeplitz linear system, where A is Hermitian or just normal. These expressions and bounds are in terms of the three parameters that define A and Chebyshev polynomials of the first or second kind. AMS subject classification (2000)  65F10, 65N22     

An efficient algorithm for regularization of Laplace transform inversion in real case

We address design of a numerical algorithm for solving the linear system arising in numerical inversion of Laplace transforms in real case [L. D’Amore, A. Murli, Regularization of a Fourier series method for the Laplace transform inversion with real data, Inverse Problems 18 (2002) 1185–1205]. The matrix has a condition number that grows almost exponentially and the singular values decay gradually towards zero. In such a case, because of this intrinsic strong instability, the main difficulty of any numerical computation is the ability of discovering at run time, only using data, what is the maximum attainable accuracy on the solution.

In this paper, we use GMRES with the aim of relating the current residuals to the maximum attainable accuracy of the approximate solution by using a suitable stopping rule. We prove that GMRES stops after, at most, as many iterations as the number of the largest eigenvalues (compared to the machine epsilon). We use a split preconditioner that symmetrically precondition the initial system. By this way, the largest eigenvalue dynamically provides the estimate of the condition number of the matrix.     

GMRES算法在二维定常无粘流计算中的应用

发展了GMRES算法的两种不同预处理方法求解二维无粘流体动力学方程组。在保证计算效率的基础上,采用了一种减小内存需求的途径。用两个算例对GMRES算法以及两种不同的预处理方法进行分析,同时与DDADI方法进行比较。通过对NACA0012有攻角超临界流动以及GAMM通道超音流的计算,表明两种预处理下的GMRES算法都具有收敛速度快的优点,LUSGS预处理方法略优于ILU预处理方法。     

三维弹性问题Taylor展开多极边界元法的误差分析

Taylor展开多极边界元法有效的提高了边界元法的求解效率,使之可用于大规模问题的计算。然而,由于计算中对基本解进行了Taylor级数展开,与传统边界元方法相比计算精度有所下降。本文主要针对三维弹性问题Taylor展开多极边界元法的计算精度和误差进行研究。文中对两种方法的计算精度进行了比较;研究了核函数的Taylor展开性质;推导了三维弹性问题基本解的误差估计公式;给出了Taylor展开多极边界元法中远近场的划分原则。通过具体的算例,证明了该方法的正确性和误差估计公式的有效性,说明了影响Taylor展开多极边界元法求解精度的因素。     

Hybrid finite element/volume method for 3D incompressible flows with heat transfer

An implicit hybrid finite element (FE)/volume solver has been extended to incompressible flows coupled with the energy equation. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centred finite volume (FV) method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centres and the auxiliary variable at vertices. The Generalized Minimal Residual (GMRES) matrix-free strategy is adapted to solve the governing equations in both FE and FV methods. The presented 2D and 3D numerical examples show the robustness and accuracy of the numerical method.     

On the Computation of Compressible Turbulent Flows on Unstructured Grids

An accurate, fast, matrix-free implicit method has been developed to solve compressible turbulent How problems using the Spalart and Allmaras one equation turbulence model on unstructured meshes. The mean-flow and turbulence-model equations are decoupled in the time integration in order to facilitate the incorporation of different turbulence models and reduce memory requirements. Both mean flow and turbulent equations are integrated in time using a linearized implicit scheme. A recently developed, fast, matrix-free implicit method, GMRES+LU-SGS, is then applied to solve the resultant system of linear equations. The spatial discretization is carried out using a hybrid finite volume and finite element method, where the finite volume approximation based on a containment dual control volume rather than the more popular median-dual control volume is used to discretize the inviscid fluxes, and the finite element approximation is used to evaluate the viscous flux terms. The developed method is used to compute a variety of turbulent flow problems in both 2D and 3D. The results obtained are in good agreement with theoretical and experimental data and indicate that the present method provides an accurate, fast, and robust algorithm for computing compressible turbulent flows on unstructured meshes.     

Analysis of the convergence of the minimal and the orthogonal residual methods

We consider two Krylov subspace methods for solving linear systems, which are the minimal residual method and the orthogonal residual method. These two methods are studied without referring to any particular implementations. By using the Petrov–Galerkin condition, we describe the residual norms of these two methods in terms of Krylov vectors, and the relationship between there two norms. We define the Ritz singular values, and prove that the convergence of these two methods is governed by the convergence of the Ritz singular values. AMS subject classification 65F10     

GMRESR: a family of nested GMRES methods

Recently Eirola and Nevanlinna have proposed an iterative solution method for unsymmetric linear systems, in which the preconditioner is updated from step to step. Following their ideas we suggest variants of GMRES, in which a preconditioner is constructed per iteration step by a suitable approximation process, e.g., by GMRES itself. Our numerical experiments indicate that this may lead to considerable savings in CPU-time and memory requirements in typical CFD applications.