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Johannes Kraus - One of the best experts on this subject based on the ideXlab platform.

  • Multigrid Methods for convection diffusion problems discretized by a monotone scheme
    Computer Methods in Applied Mechanics and Engineering, 2017
    Co-Authors: Johannes Kraus, N R Bayramov
    Abstract:

    Abstract We study Multigrid (MG) Methods for the solution of systems of linear algebraic equations obtained from a stable discretization of convection–diffusion problems by an exponential fitting scheme. The latter ensures the stability of the simplest possible coarse grid operators obtained from Galerkin projections based on graph matching. Linear and nonlinear MG preconditioners are defined in the framework of algebraic multilevel iteration. The option of using polynomial smoothers is investigated in context of nonsymmetric problems and a systematic performance comparison is presented for various algorithms on a representative set of two- and three-dimensional test problems.

  • Multigrid Methods for isogeometric discretization.
    Computer Methods in Applied Mechanics and Engineering, 2013
    Co-Authors: Krishan Pratap Singh Gahalaut, Johannes Kraus, Satyendra Tomar
    Abstract:

    Abstract We present (geometric) Multigrid Methods for isogeometric discretization of scalar second order elliptic problems. The smoothing property of the relaxation method, and the approximation property of the intergrid transfer operators are analyzed. These properties, when used in the framework of classical Multigrid theory, imply uniform convergence of two-grid and Multigrid Methods. Supporting numerical results are provided for the smoothing property, the approximation property, convergence factor and iterations count for V -, W - and F -cycles, and the linear dependence of V -cycle convergence on the smoothing steps. For two dimensions, numerical results include the problems with variable coefficients, simple multi-patch geometry, a quarter annulus, and the dependence of convergence behavior on refinement levels l , whereas for three dimensions, only the constant coefficient problem in a unit cube is considered. The numerical results are complete up to polynomial order p = 4 , and for C 0 and C p - 1 smoothness.

N R Bayramov - One of the best experts on this subject based on the ideXlab platform.

  • Multigrid Methods for convection diffusion problems discretized by a monotone scheme
    Computer Methods in Applied Mechanics and Engineering, 2017
    Co-Authors: Johannes Kraus, N R Bayramov
    Abstract:

    Abstract We study Multigrid (MG) Methods for the solution of systems of linear algebraic equations obtained from a stable discretization of convection–diffusion problems by an exponential fitting scheme. The latter ensures the stability of the simplest possible coarse grid operators obtained from Galerkin projections based on graph matching. Linear and nonlinear MG preconditioners are defined in the framework of algebraic multilevel iteration. The option of using polynomial smoothers is investigated in context of nonsymmetric problems and a systematic performance comparison is presented for various algorithms on a representative set of two- and three-dimensional test problems.

Jun Zhang - One of the best experts on this subject based on the ideXlab platform.

Ludmil T. Zikatanov - One of the best experts on this subject based on the ideXlab platform.

  • Algebraic Multigrid Methods
    Acta Numerica, 2017
    Co-Authors: Jinchao Xu, Ludmil T. Zikatanov
    Abstract:

    This paper provides an overview of AMG Methods for solving large-scale systems of equations, such as those from discretizations of partial differential equations. AMG is often understood as the acronym of ‘algebraic Multigrid’, but it can also be understood as ‘abstract Multigrid’. Indeed, we demonstrate in this paper how and why an algebraic Multigrid method can be better understood at a more abstract level. In the literature, there are many different algebraic Multigrid Methods that have been developed from different perspectives. In this paper we try to develop a unified framework and theory that can be used to derive and analyse different algebraic Multigrid Methods in a coherent manner. Given a smoother $R$ for a matrix $A$ , such as Gauss–Seidel or Jacobi, we prove that the optimal coarse space of dimension $n_{c}$ is the span of the eigenvectors corresponding to the first $n_{c}$ eigenvectors $\bar{R}A$ (with $\bar{R}=R+R^{T}-R^{T}AR$ ). We also prove that this optimal coarse space can be obtained via a constrained trace-minimization problem for a matrix associated with $\bar{R}A$ , and demonstrate that coarse spaces of most existing AMG Methods can be viewed as approximate solutions of this trace-minimization problem. Furthermore, we provide a general approach to the construction of quasi-optimal coarse spaces, and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation and the anisotropy. Our theory applies to most existing Multigrid Methods, including the standard geometric Multigrid method, classical AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG and spectral AMGe.

  • Algebraic Multigrid Methods
    arXiv: Numerical Analysis, 2016
    Co-Authors: Jinchao Xu, Ludmil T. Zikatanov
    Abstract:

    This paper is to give an overview of AMG Methods for solving large scale systems of equations such as those from the discretization of partial differential equations. AMG is often understood as the acronym of "Algebraic Multi-Grid", but it can also be understood as "Abstract Muti-Grid". Indeed, as it demonstrates in this paper, how and why an algebraic Multigrid method can be better understood in a more abstract level. In the literature, there are a variety of different algebraic Multigrid Methods that have been developed from different perspectives. In this paper, we try to develop a unified framework and theory that can be used to derive and analyze different algebraic Multigrid Methods in a coherent manner. Given a smoother $R$ for a matrix $A$, such as Gauss-Seidel or Jacobi, we prove that the optimal coarse space of dimension $n_c$ is the span of the eigen-vectors corresponding to the first $n_c$ eigenvalues of $\bar RA$ (with $\bar R=R+R^T-R^TAR$). We also prove that this optimal coarse space can be obtained by a constrained trace-minimization problem for a matrix associated with $\bar RA$ and demonstrate that coarse spaces of most of existing AMG Methods can be viewed some approximate solution of this trace-minimization problem. Furthermore, we provide a general approach to the construction of a quasi-optimal coarse space and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation, and the anisotropy. Our theory applies to most existing Multigrid Methods, including the standard geometric Multigrid method, the classic AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG, and spectral AMGe.

  • algebraic Multigrid Methods based on compatible relaxation and energy minimization
    2007
    Co-Authors: James Brannick, Ludmil T. Zikatanov
    Abstract:

    This paper presents an adaptive algebraic Multigrid method for the solution of positive definite linear systems arising from the discretizations of elliptic partial differential equations. The proposed method uses compatible relaxation to adaptively construct the set of coarse variables. The nonzero supports for the coarse-space basis is determined by approximation of the so-called two-level “ideal” interpolation operator. Then, an energy minimizing coarse basis is formed using an approach aimed to minimize the trace of the coarse-level operator. The presented approach maintains Multigrid-like optimality, without the need for parameter tuning, for some problems where current algorithms exhibit degraded performance. Numerical experiments are presented that demonstrate the efficacy of the approach.

  • On an energy minimizing basis for algebraic Multigrid Methods
    Computing and Visualization in Science, 2004
    Co-Authors: Ludmil T. Zikatanov
    Abstract:

    This paper is devoted to the study of an energy minimizing basis first introduced in Wan, Chan and Smith (2000) for algebraic Multigrid Methods. The basis will be first obtained in an explicit and compact form in terms of certain local and global operators. The basis functions are then proved to be locally harmonic functions on each coarse grid "element". Using these new results, it is illustrated that this basis can be numerically obtained in an optimal fashion. In addition to the intended application for algebraic Multigrid method, the energy minimizing basis may also be applied for numerical homogenization.

  • uniformly convergent Multigrid Methods for convection diffusion problems without any constraint on coarse grids
    Advances in Computational Mathematics, 2004
    Co-Authors: Hwanho Kim, Ludmil T. Zikatanov
    Abstract:

    We construct a class of Multigrid Methods for convection–diffusion problems. The proposed algorithms use first order stable monotone schemes to precondition the second order standard Galerkin finite element discretization. To speed up the solution process of the lower order schemes, cross-wind-block reordering of the unknowns is applied. A V-cycle iteration, based on these algorithms, is then used as a preconditioner in GMRES. The numerical examples show that this method is convergent without imposing any constraint on the coarsest grid and the convergence of the preconditioned method is uniform.

Walter Zulehner - One of the best experts on this subject based on the ideXlab platform.

  • numerical simulation of fluid structure interaction problems on hybrid meshes with algebraic Multigrid Methods
    Journal of Computational and Applied Mathematics, 2011
    Co-Authors: Huidong Yang, Walter Zulehner
    Abstract:

    Fluid-structure interaction problems arise in many fields of application such as flows around elastic structures and blood flow in arteries. The method presented in this paper for solving such a problem is based on a reduction to an equation at the interface, involving the so-called Steklov-Poincare operators. This interface equation is solved by a Newton iteration, for which directional derivatives involving shape derivatives with respect to the interface perturbation have to be evaluated appropriately. One step of the Newton iteration requires the solution of several decoupled linear sub-problems in the structure and the fluid domains. These sub-problems are spatially discretized by a finite element method on hybrid meshes. For the time discretization, implicit first-order Methods are used for both sub-problems. The discretized equations are solved by algebraic Multigrid Methods.

  • Convergence analysis of Multigrid Methods with collective point smoothers for optimal control problems
    Computing and Visualization in Science, 2011
    Co-Authors: Stefan Takacs, Walter Zulehner
    Abstract:

    In this paper we consider Multigrid Methods for solving saddle point problems. The choice of an appropriate smoothing strategy is a key issue in this case. Here we focus on the widely used class of collective point smoothers. These Methods are constructed by a point-wise grouping of the unknowns leading to, e.g., collective Richardson, Jacobi or Gauss-Seidel relaxation Methods. Their smoothing properties are well-understood for scalar problems in the symmetric and positive definite case. In this work the analysis of these Methods is extended to a special class of saddle point problems, namely to the optimality system of optimal control problems. For elliptic distributed control problems we show that the convergence rates of Multigrid Methods with collective point smoothers are bounded independent of the grid size and the regularization (or cost) parameter.

  • numerical simulation of fluid structure interaction problems on hybrid meshes with algebraic Multigrid Methods
    International Conference on Large-Scale Scientific Computing, 2009
    Co-Authors: Huidong Yang, Walter Zulehner
    Abstract:

    Fluid-structure interaction problems arise in many application fields such as flows around elastic structures or blood flow problems in arteries The method presented in this paper for solving such a problem is based on a reduction to an equation at the interface, involving the so-called Steklov-Poincare operators This interface equation is solved by a Newton-like iteration One step of the Newton-like iteration requires the solution of several decoupled linear subproblems in the structural and the fluid domains These subproblems are spatially discretized by a finite element method on hybrid meshes For the time discretization implicit first-order Methods are used for both subproblems The discretized equations are solved by algebraic Multigrid Methods.