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Robust parameterfree algebraic multilevel preconditioning
"... To precondition large sparse linear systems resulting from the discretization of secondorder elliptic partial di erential equations, many recent works focus on the socalled algebraic multilevel methods. These are based on a block incomplete factorization process applied to the system matrix partit ..."
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To precondition large sparse linear systems resulting from the discretization of secondorder elliptic partial di erential equations, many recent works focus on the socalled algebraic multilevel methods. These are based on a block incomplete factorization process applied to the system matrix partitioned in hierarchical form. They have been shown to be both robust and e cient in several circumstances, leading to iterative solution schemes of optimal order of computational complexity. Now, despite the procedure is essentially algebraic, previous works focus generally on a speci c context and consider schemes that use classical grid hierarchies with characteristic mesh sizes h; 2h; 4h, etc. Therefore, these methods require some extra information besides the matrix of the linear system and lack of robustness in some situations where semicoarsening would be desirable. In this paper, we develop a general method that can be applied in a black box fashion to a wide class of problems, ranging from 2D model Poisson problems to 3D singularly perturbed convection–di usion equations. It is based on an automatic coarsening process similar to the one used in the AMG method, and on coarse grid matrices computed according to a simple and cheap aggregation principle. Numerical experiments illustrate the e ciency and the robustness of the proposed approach. Copyright? 2002 John Wiley & Sons, Ltd. KEY WORDS: iterative methods; convergence; preconditioning
A robust algebraic multilevel preconditioner for nonsymmetric Mmatrices
, 2000
"... Stable finite difference approximations of convection–diffusion equations lead to large sparse linear systems of equations whose coefficient matrix is an Mmatrix, which is highly nonsymmetric when the convection dominates. For an efficient iterative solution of such systems, it is proposed to cons ..."
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Cited by 8 (5 self)
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Stable finite difference approximations of convection–diffusion equations lead to large sparse linear systems of equations whose coefficient matrix is an Mmatrix, which is highly nonsymmetric when the convection dominates. For an efficient iterative solution of such systems, it is proposed to consider in the nonsymmetric case an algebraic multilevel preconditioning method formerly proposed for pure diffusion problems, and for which theoretical results prove grid independent convergence in this context. These results are supplemented here by a Fourier analysis that applies to constant coefficient problems with periodic boundary conditions whenever using an ‘idealized ’ version of the twolevel preconditioner. Within this setting, it is proved that any eigenvalue λ of the preconditioned system satisfies � �λ−1 − 1 − ic � � ≤ 1 1 2 for some real constant c such that c  ≤ 4. This result holds independently of the grid size and uniformly with respect to the ratio between convection and diffusion. Extensive numerical experiments are conducted to assess the convergence of practical two and multilevel schemes. These experiments, which include problems with highly variable and rotating convective flow, indicate that the convergence is grid independent. It deteriorates moderately as the convection becomes increasingly dominating, but the convergence factor remains uniformly bounded. This conclusion is supported for both uniform and some nonuniform (stretched) grids.
A Robust Algebraic Preconditioner for Finite Difference Approximations of ConvectionDiffusion Equations
, 1999
"... Stable finite difference approximations of convectiondiffusion equations lead to large sparse linear systems of equations whose coefficient matrix is an Mmatrix, which is highly non symmetric when the convection dominates. For an efficient iterative solution of such systems, it is proposed to con ..."
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Stable finite difference approximations of convectiondiffusion equations lead to large sparse linear systems of equations whose coefficient matrix is an Mmatrix, which is highly non symmetric when the convection dominates. For an efficient iterative solution of such systems, it is proposed to consider in the non symmetric case an algebraic multilevel preconditioning method formerly proposed for pure diffusion problems, and for which theoretical results prove grid independent convergence in this context. These results are supplemented here by a Fourier analysis which applies to constant coefficient problems with periodic boundary conditions whenever using an "idealized" version of the twolevel preconditioner. Within this setting, it is proved that any eigenvalue of the preconditioned system satisfies j \Gamma1 \Gamma 1 \Gamma i cj 1 2 for some real constant c such that jcj 1 4 . This result holds independently of the grid size and uniformly with respect to the ratio betwee...
On a modification of algebraic multilevel iteration method for finite element matrices *
"... A M S subject classification: 65F10, 65N20 Ларин М.Р. О модификации алгебраического многоуровневого итерационного метода для решения конечноэлементных систем линейных алгебраических уравнений // Сиб. журн. вычисл. математики / РАН. Сиб. отдние.Новосибирск, 2007.Т. 10, № 1.С. 101116. В нас ..."
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A M S subject classification: 65F10, 65N20 Ларин М.Р. О модификации алгебраического многоуровневого итерационного метода для решения конечноэлементных систем линейных алгебраических уравнений // Сиб. журн. вычисл. математики / РАН. Сиб. отдние.Новосибирск, 2007.Т. 10, № 1.С. 101116. В настоящее время многосеточные и многоуровневые методы очень популярны для решения разреженных систем линейных алгебраических уравнений. Они обладают как широкой областью применения, так и эффективностью. В работе [6] был предложен алгебраический многоуровневый итерационный (AMLI) метод для решения конечноэлементных систем линейных алгебраических уравнений. Однако этот метод имеет два ограничения на свойства исходной матрицы, которые могут нарушаться на практике. C целью избежать их и улучшить качество AMLIпредбуславливателя предлагается и анализируется семейство итерационных параметров релаксации. Ключевые слова: алгебраический многоуровневый метод, метод сопряженных градиентов с предобуславливателем, системы линейных алгебраических уравнений, метод конечных элементов. Today, multigrids and multilevel methods for solving a sparse linear system of equations are well known. They are both robust and efficient. In