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Multigrid Methods (multigrid + methods)
Selected AbstractsPositive-definite q -families of continuous subcell Darcy-flux CVD(MPFA) finite-volume schemes and the mixed finite element methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2008Michael G. Edwards Abstract A new family of locally conservative cell-centred flux-continuous schemes is presented for solving the porous media general-tensor pressure equation. A general geometry-permeability tensor approximation is introduced that is piecewise constant over the subcells of the control volumes and ensures that the local discrete general tensor is elliptic. A family of control-volume distributed subcell flux-continuous schemes are defined in terms of the quadrature parametrization q (Multigrid Methods. Birkhauser: Basel, 1993; Proceedings of the 4th European Conference on the Mathematics of Oil Recovery, Norway, June 1994; Comput. Geosci. 1998; 2:259,290), where the local position of flux continuity defines the quadrature point and each particular scheme. The subcell tensor approximation ensures that a symmetric positive-definite (SPD) discretization matrix is obtained for the base member (q=1) of the formulation. The physical-space schemes are shown to be non-symmetric for general quadrilateral cells. Conditions for discrete ellipticity of the non-symmetric schemes are derived with respect to the local symmetric part of the tensor. The relationship with the mixed finite element method is given for both the physical-space and subcell-space q -families of schemes. M -matrix monotonicity conditions for these schemes are summarized. A numerical convergence study of the schemes shows that while the physical-space schemes are the most accurate, the subcell tensor approximation reduces solution errors when compared with earlier cell-wise constant tensor schemes and that subcell tensor approximation using the control-volume face geometry yields the best SPD scheme results. A particular quadrature point is found to improve numerical convergence of the subcell schemes for the cases tested. Copyright © 2007 John Wiley & Sons, Ltd. [source] Multigrid methods for the symmetric interior penalty method on graded meshesNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6 2009S. C. Brenner Abstract The symmetric interior penalty (SIP) method on graded meshes and its fast solution by multigrid methods are studied in this paper. We obtain quasi-optimal error estimates in both the energy norm and the L2 norm for the SIP method, and prove uniform convergence of the W -cycle multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical results. Copyright © 2009 John Wiley & Sons, Ltd. [source] FLEXMG: A new library of multigrid preconditioners for a spectral/finite element incompressible flow solverINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2010M. Rasquin Abstract A new library called FLEXMG has been developed for a spectral/finite element incompressible flow solver called SFELES. FLEXMG allows the use of various types of iterative solvers preconditioned by algebraic multigrid methods. Two families of algebraic multigrid preconditioners have been implemented, namely smooth aggregation-type and non-nested finite element-type. Unlike pure gridless multigrid, both of these families use the information contained in the initial fine mesh. A hierarchy of coarse meshes is also needed for the non-nested finite element-type multigrid so that our approaches can be considered as hybrid. Our aggregation-type multigrid is smoothed with either a constant or a linear least-square fitting function, whereas the non-nested finite element-type multigrid is already smooth by construction. All these multigrid preconditioners are tested as stand-alone solvers or coupled with a GMRES method. After analyzing the accuracy of the solutions obtained with our solvers on a typical test case in fluid mechanics, their performance in terms of convergence rate, computational speed and memory consumption is compared with the performance of a direct sparse LU solver as a reference. Finally, the importance of using smooth interpolation operators is also underlined in the study. Copyright © 2010 John Wiley & Sons, Ltd. [source] Efficient preconditioners for boundary element matrices based on grey-box algebraic multigrid methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2003U. Langer Abstract This paper is concerned with the iterative solution of the boundary element equations arising from standard Galerkin boundary element discretizations of first-kind boundary integral operators of positive and negative order. We construct efficient preconditioners on the basis of so-called grey-box algebraic multigrid methods that are well adapted to the treatment of boundary element matrices. In particular, the coarsening is based on an auxiliary matrix that represents the underlying topology in a certain sense. This auxiliary matrix is additionally used for the construction of the smoothers and the transfer operators. Finally, we present the results of some numerical studies that show the efficiency of the proposed algebraic multigrid preconditioners. Copyright © 2003 John Wiley & Sons, Ltd. [source] Evaluation of three unstructured multigrid methods on 3D finite element problems in solid mechanics,INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2002Mark Adams Abstract Multigrid has been a popular solver method for finite element and finite difference problems with regular grids for over 20 years. The application of multigrid to unstructured grid problems, in which it is often difficult or impossible for the application to provide coarse grids, is not as well understood. In particular, methods that are designed to require only data that are easily available in most finite element applications (i.e. fine grid data), constructing the grid transfer operators and coarse grid operators internally, are of practical interest. We investigate three unstructured multigrid methods that show promise for challenging problems in 3D elasticity: (1) non-nested geometric multigrid, (2) smoothed aggregation, and (3) plain aggregation algebraic multigrid. This paper evaluates the effectiveness of these three methods on several unstructured grid problems in 3D elasticity with up to 76 million degrees of freedom. Published in 2002 by John Wiley & Sons, Ltd. [source] BILU implicit multiblock Euler/Navier,Stokes simulation for rotor tip vortex and wake convectionINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2007Bowen Zhong Abstract In this paper, a block incomplete lower,upper (BILU) decomposition method is incorporated with a multiblock three-dimensional Euler/Navier,Stokes solver for simulation of hovering rotor tip vortices and rotor wake convection. Results of both Euler and Navier,Stokes simulations are obtained and compared with experimental observations. The comparisons include surface pressure distributions and tip vortex trajectories. The comparisons suggest that resolution of the boundary layer is important for the accurate evaluation of the blade surface loading, but is less so for the correct prediction of the vortex trajectory. Numerical tests show that, using Courant,Friedrichs,Lewy (CFL) number of 10 or 30 with the developed BILU implicit scheme can be 6,7 times faster than an explicit scheme. The importance of solution acceleration schemes that increase the permitted time-step is illustrated by comparing the evolving wake structures at different stages of the calculation. In contrast to fixed wing simulations, the extent of the wake structures is shown to require resolution of large physical time. This observation explains the poor performance that is obtained when employing convergence acceleration strategies originally intended for solution of equilibrium problems, such as the multigrid methods. Copyright © 2007 John Wiley & Sons, Ltd. [source] Higher order finite element methods and multigrid solvers in a benchmark problem for the 3D Navier,Stokes equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2002Volker John Abstract This paper presents a numerical study of the 3D flow around a cylinder which was defined as a benchmark problem for the steady state Navier,Stokes equations within the DFG high-priority research program flow simulation with high-performance computers by Schafer and Turek (Vol. 52, Vieweg: Braunschweig, 1996). The first part of the study is a comparison of several finite element discretizations with respect to the accuracy of the computed benchmark parameters. It turns out that boundary fitted higher order finite element methods are in general most accurate. Our numerical study improves the hitherto existing reference values for the benchmark parameters considerably. The second part of the study deals with efficient and robust solvers for the discrete saddle point problems. All considered solvers are based on coupled multigrid methods. The flexible GMRES method with a multiple discretization multigrid method proves to be the best solver. Copyright © 2002 John Wiley & Sons, Ltd. [source] An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matricesNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2010M. Donatelli Abstract Local Fourier analysis (LFA) is a classical tool for proving convergence theorems for multigrid methods (MGMs). In particular, we are interested in optimal convergence, i.e. convergence rates that are independent of the problem size. For elliptic partial differential equations (PDEs), a well-known optimality result requires that the sum of the orders of the grid transfer operators is not lower than the order of the PDE approximated. Analogously, when dealing with MGMs for Toeplitz matrices, a well-known optimality condition concerns the position and the order of the zeros of the symbols of the grid transfer operators. In this work we show that in the case of elliptic PDEs with constant coefficients, the two different approaches lead to an equivalent condition. We argue that the analysis for Toeplitz matrices is an algebraic generalization of the LFA, which allows to deal not only with differential problems but also for instance with integral problems. The equivalence of the two approaches gives the possibility of using grid transfer operators with different orders also for MGMs for Toeplitz matrices. We give also a class of grid transfer operators related to the B-spline's refinement equation and study their geometric properties. Numerical experiments confirm the correctness of the proposed analysis. Copyright © 2010 John Wiley & Sons, Ltd. [source] Performance of algebraic multigrid methods for non-symmetric matrices arising in particle methodsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2010B. Seibold Abstract Large linear systems with sparse, non-symmetric matrices are known to arise in the modeling of Markov chains or in the discretization of convection,diffusion problems. Due to their potential of solving sparse linear systems with an effort that is linear in the number of unknowns, algebraic multigrid (AMG) methods are of fundamental interest for such systems. For symmetric positive definite matrices, fundamental theoretical convergence results are established, and efficient AMG solvers have been developed. In contrast, for non-symmetric matrices, theoretical convergence results have been provided only recently. A property that is sufficient for convergence is that the matrix be an M-matrix. In this paper, we present how the simulation of incompressible fluid flows with particle methods leads to large linear systems with sparse, non-symmetric matrices. In each time step, the Poisson equation is approximated by meshfree finite differences. While traditional least squares approaches do not guarantee an M-matrix structure, an approach based on linear optimization yields optimally sparse M-matrices. For both types of discretization approaches, we investigate the performance of a classical AMG method, as well as an algebraic multilevel iteration (AMLI) type method. While in the considered test problems, the M-matrix structure turns out not to be necessary for the convergence of AMG, problems can occur when it is violated. In addition, the matrices obtained by the linear optimization approach result in fast solution times due to their optimal sparsity. Copyright © 2010 John Wiley & Sons, Ltd. [source] Multigrid methods for the symmetric interior penalty method on graded meshesNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6 2009S. C. Brenner Abstract The symmetric interior penalty (SIP) method on graded meshes and its fast solution by multigrid methods are studied in this paper. We obtain quasi-optimal error estimates in both the energy norm and the L2 norm for the SIP method, and prove uniform convergence of the W -cycle multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical results. Copyright © 2009 John Wiley & Sons, Ltd. [source] ,-matrix preconditioners for symmetric saddle-point systems from meshfree discretizationNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 10 2008Sabine Le Borne Abstract Meshfree methods are suitable for solving problems on irregular domains, avoiding the use of a mesh. To deal with the boundary conditions, we can use Lagrange multipliers and obtain a sparse, symmetric and indefinite system of saddle-point type. Many methods have been developed to solve the indefinite system. Previously, we presented an algebraic method to construct an LU-based preconditioner for the saddle-point system obtained by meshfree methods, which combines the multilevel clustering method with the ,-matrix arithmetic. The corresponding preconditioner has both ,-matrix and sparse matrix subblocks. In this paper we refine the above method and propose a way to construct a pure ,-matrix preconditioner. We compare the new method with the old method, JOR and smoothed algebraic multigrid methods. The numerical results show that the new preconditioner outperforms the preconditioners based on the other methods. Copyright © 2008 John Wiley & Sons, Ltd. [source] On algebraic multigrid methods derived from partition of unity nodal prolongatorsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2006Tim Boonen Abstract This paper is concerned with algebraic multigrid for finite element discretizations of the divgrad, curlcurl and graddiv equations on tetrahedral meshes with piecewise linear shape functions. First, an edge, face and volume prolongator are derived from an arbitrary partition of unity nodal prolongator for a tetrahedral fine mesh, using the formulas for edge, face and volume elements. This procedure can be repeated recursively. The implied coarse topology and the normalization of the prolongators are analysed. It is proved that the range spaces of the nodal prolongator and of the derived edge, face and volume prolongators form a discrete de Rham complex if these prolongators have full rank. It is shown that on simplicial meshes, the constructed edge prolongator is a generalization of the Reitzinger,Schöberl prolongator. The derived edge and face prolongators are applied in an algebraic multigrid method for the curlcurl and graddiv equations, and numerical results are presented. Copyright © 2006 John Wiley & Sons, Ltd. [source] A systematic comparison of coupled and distributive smoothing in multigrid for the poroelasticity systemNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2004F. J. Gaspar Abstract In this paper, we present efficient multigrid methods for the system of poroelasticity equations discretized on a staggered grid. In particular, we compare two different smoothing approaches with respect to efficiency and robustness. One approach is based on the coupled relaxation philosophy. We introduce ,cell-wise' and ,line-wise' versions of the coupled smoothers. They are compared with a distributive relaxation, that gives us a decoupled system of equations. It can be smoothed equation-wise with basic iterative methods. All smoothing methods are evaluated for the same poroelasticity test problems in which parameters, like the time step, or the Lamé coefficients are varied. Some highly efficient methods result, as is confirmed by the numerical experiments. Copyright © 2004 John Wiley & Sons, Ltd. [source] Multigrid approaches to non-linear diffusion problems on unstructured meshesNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 8 2001Dimitri J. Mavriplis Abstract The efficiency of three multigrid methods for solving highly non-linear diffusion problems on two-dimensional unstructured meshes is examined. The three multigrid methods differ mainly in the manner in which the non-linearities of the governing equations are handled. These comprise a non-linear full approximation storage (FAS) multigrid method which is used to solve the non-linear equations directly, a linear multigrid method which is used to solve the linear system arising from a Newton linearization of the non-linear system, and a hybrid scheme which is based on a non-linear FAS multigrid scheme, but employs a linear solver on each level as a smoother. Results indicate that, in the asymptotic convergence region, all methods are equally effective at converging the non-linear residual in a given number of multigrid cycles, but that the linear solver is more efficient in cpu time due to the lower cost of linear versus non-linear grid sweeps. Copyright © 2001 John Wiley & Sons, Ltd. [source] A multigrid upwind strategy for accelerating steady-state computations of waves propagating with curvature-dependent speedsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2002Jonathan Rochez Abstract A multigrid strategy using upwind finite differencing is developed for accelerating the steady state computations of waves, [14] propagating with curvature-dependent speeds. This will allow the rapid computation of a "burn table." In a high explosive material, a burn table will allow the elimination of solving chemical reaction ODEs by feeding in source terms to the reactive flow equations for solution of the system of ignition of the high explosive material. Standard iterative methods show a quick reduction of the residual followed by a slow final convergence to the solution at high iterations. Such systems, including a nonlinear system such as this, are excellent choices for the use of multigrid methods to speed up convergence. Numerical steady-state solutions to the eikonal equation on several test grids are conducted. Results are presented for these cases in 2D and a cubic grid in 3D using a Runge-Kutta time iteration for the smoothing operator until steady state is reached. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 179,192, 2002; DOI 10.1002/num.1002 [source] | |||||||||||||