A generalized BPX multigrid framework covering nonnested V-cycle methods

More than a decade ago, Bramble, Pasciak and Xu developed a framework in analyzing the multigrid methods with nonnested spaces or noninherited quadratic forms. It was subsequently known as the BPX multigrid framework, which was widely used in the analysis of multigrid and domain decomposition methods. However, the framework has an apparent limit in the analysis of nonnested V-cycle methods, and it produces a variable V-cycle, or nonuniform convergence rate V-cycle methods, or other nonoptimal results in analysis thus far.

This paper completes a long-time effort in extending the BPX multigrid framework so that it truly covers the nonnested V-cycle. We will apply the extended BPX framework to the analysis of many V-cycle nonnested multigrid methods. Some of them were proven previously only for two-level and W-cycle iterations. Some numerical results are presented to support the theoretical analysis of this paper.

     

MORTAR型旋转Q_1元的多重网格方法

本文讨论了mortar型旋转Q_1元的多重网格方法.证明了W循环的多重网格法是最优的,即收敛率与网格尺寸及层数无关.同时给出了一种可变的V循环多重网格算法,得到了一个条件数一致有界的预条件子.最后,数值试验验证了我们的理论结果.     

MULTIGRID FOR THE MORTAR ELEMENT METHOD WITH LOCALLY P_1 NONCONFORMING ELEMENTS

In this paper we study the theoretical properties of multigrid algorithm for discretization of the Poisson equation in 2D using a mortar element method under the assumption that the triangulations on every subdomain are uniform. We prove the convergence of the W-cycle with a sufficiently large number of smoothing steps. The     

A V-CYCLE MULTIGRID METHOD FOR THE PLATE BENDINGPROBLEM DISCRETIZED BY NONCONFORMING FINITEELEMENTS

1.IntroductionMultigridmethodshavebecomesomeofthemostpowerfulmethodsforsolvingpartialdifferentialequationsdiscretizedbythefiniteelementandfinitedifferencemethods.(of.[7][11][141andreferencetherein).Multigridmethodsforthenonconformingfiniteelementshav...     

The analysis of multigrid algorithms for cell centered finite difference methods

In this paper, we examine multigrid algorithms for cell centered finite difference approximations of second order elliptic boundary value problems. The cell centered application gives rise to one of the simplest non-variational multigrid algorithms. We shall provide an analysis which guarantees that the W-cycle and variable V-cycle multigrid algorithms converge with a rate of iterative convergence which can be bounded independently of the number of multilevel spaces. In contrast, the natural variational multigrid algorithm converges much more slowly.     

Multigrid for the mortar element method for P1 nonconforming element

In this paper, a multigrid algorithm is presented for the mortar element method for P1 nonconforming element. Based on the theory developed by Bramble, Pasciak, Xu in [5], we prove that the W-cycle multigrid is optimal, i.e. the convergence rate is independent of the mesh size and mesh level. Meanwhile, a variable V-cycle multigrid preconditioner is constructed, which results in a preconditioned system with uniformly bounded condition number. Received May 11, 1999 / Revised version received April 1, 2000 / Published online October 16, 2000     

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations

We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell's equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their "immediate" neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.     

AN OPTIMAL V-CYCLE MULTIGRID METHOD FOR CONFORMING AND NONCONFORMING PLATE ELEMENTS

In this paper, an optimal V-cycle multigrid method for some conforming and nonconforming plate elements are constructed. A new method dealing with nonnested multigrid methods is presented.     

Uniform convergence of the multigrid V-cycle for an anisotropic problem

In this paper, we consider the linear systems arising from the standard finite element discretizations of certain second order anisotropic problems with variable coefficients on a rectangle. We study the performance of a V-cycle multigrid method applied to the finite element equations. Since the usual ``regularity and approximation' assumption does not hold for the anisotropic finite element problems, the standard multigrid convergence theory cannot be applied directly. In this paper, a modification of the theory of Braess and Hackbusch will be presented. We show that the V-cycle multigrid iteration with a line smoother is a uniform contraction in the energy norm. In the verification of the hypotheses in our theory, we use a weighted -norm estimate for the error in the Galerkin finite element approximation and a smoothing property of the line smoothers which is proved in this paper.

     

Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems

In this paper we prove the uniform convergence of the standard multigrid V-cycle algorithm with the Gauss-Seidel relaxation performed only on the new nodes and their "immediate" neighbors for discrete elliptic problems on the adaptively refined finite element meshes using the newest vertex bisection algorithm. The proof depends on sharp estimates on the relationship of local mesh sizes and a new stability estimate for the space decomposition based on the Scott-Zhang interpolation operator. Extensive numerical results are reported, which confirm the theoretical analysis.     

A V-CYCLE MUTIGRID FOR QUADRILATERAL ROTATED Q1 ELEMENT WITH NUMERICAL INTEGRATION

In this paper, a V-cycle multigrid method is presented for quadrilateral rotated Q1 elements with numerical integration.     

MULTIGRID ALGORITHM FOR THE COUPLING SYSTEM OF NATURAL BOUNDARY ELEMENT METHOD AND FINITE ELEMENT METHOD FOR UNBOUNDED DOMAIN PROBLEMS

In this paper, some V-cycle multigrid algorithms are presented for the coupling system arising from the discretization of the Dirichlet exterior problem by coupling the natural boundary element method and finite element method. The convergence of these multigrid algorithms is obtained even with only one smoothing on all levels. The rate of convergence is found uniformly bounded independent of the number of levels and the mesh sizes of all levels, which indicates that these multigrid algorithms are optimal. Some numerical results are also reported.     

V-cycle multigrid methods for Wilson nonconforming element

Here two types of optimal V-cycle multigrid algorithms are presented for Wilson nonconforming finite element.     

Analysis of tensor product multigrid

We consider anisotropic second order elliptic boundary value problems in two dimensions, for which the anisotropy is exactly aligned with the coordinate axes. This includes cases where the operator features a singular perturbation in one coordinate direction, whereas its restriction to the other direction remains neatly elliptic. Most prominently, such a situation arises when polar coordinates are introduced.The common multigrid approach to such problems relies on line relaxation in the direction of the singular perturbation combined with semi-coarsening in the other direction. Taking the idea from classical Fourier analysis of multigrid, we employ eigenspace techniques to separate the coordinate directions. Thus, convergence of the multigrid method can be examined by looking at one-dimensional operators only. In a tensor product Galerkin setting, this makes it possible to confirm that the convergence rates of the multigrid V-cycle are bounded independently of the number of grid levels involved. In addition, the estimates reveal that convergence is also robust with respect to a singular perturbation in one coordinate direction.Finally, we supply numerical evidence that the algorithm performs satisfactorily in settings more general than those covered by the proof.     

V-cycle convergence of some multigrid methods for ill-posed problems

For ill-posed linear operator equations we consider some V-cycle multigrid approaches, that, in the framework of Bramble, Pasciak, Wang, and Xu (1991), we prove to yield level independent contraction factor estimates. Consequently, we can incorporate these multigrid operators in a full multigrid method, that, together with a discrepancy principle, is shown to act as an iterative regularization method for the underlying infinite-dimensional ill-posed problem. Numerical experiments illustrate the theoretical results.

     

Monotone multigrid methods for elliptic variational inequalities II

Summary. We derive globally convergent multigrid methods for discrete elliptic variational inequalities of the second kind as obtained from the approximation of related continuous problems by piecewise linear finite elements. The coarse grid corrections are computed from certain obstacle problems. The actual constraints are fixed by the preceding nonlinear fine grid smoothing. This new approach allows the implementation as a classical V-cycle and preserves the usual multigrid efficiency. We give estimates for the asymptotic convergence rates. The numerical results indicate a significant improvement as compared with previous multigrid approaches. Received March 26, 1994 / Revised version received September 22, 1994     

Optimized partial semicoarsening multigrid algorithm for heat diffusion problems and anisotropic grids

The purpose of this work is to reduce the CPU time necessary to solve three two-dimensional linear diffusive problems governed by Laplace and Poisson equations, discretized with anisotropic grids. The Finite Difference Method is used to discretizate the differential equations with central differencing scheme. The systems of equations are solved with the lexicographic and red–black Gauss–Seidel methods associated to the geometric multigrid with correction scheme and V-cycle. The anisotropic grids considered have aspect ratios varying from 1/1024 up to 16,384. Four algorithms are compared: full coarsening, semicoarsening, full coarsening followed by semicoarsening and partial semicoarsening. Three new restriction schemes for anisotropic grids are proposed: geometric half weighting, geometric full weighting and partial weighting. Comparisons are made among these three new schemes and some restriction schemes presented in literature: injection, half weighting and full weighting. The prolongation process used is the bilinear interpolation. It is also investigated the effects on the CPU time caused by: the number of inner iterations of the smoother, the number of grids and the number of grid elements. It was verified that the partial semicoarsening algorithm is the fastest. This work also provides the optimum values of the multigrid components for this algorithm.     

The convergence of V-cycle multigrid algorithms for axisymmetric Laplace and Maxwell equations

We investigate some simple finite element discretizations for the axisymmetric Laplace equation and the azimuthal component of the axisymmetric Maxwell equations as well as multigrid algorithms for these discretizations. Our analysis is targeted at simple model problems and our main result is that the standard V-cycle with point smoothing converges at a rate independent of the number of unknowns. This is contrary to suggestions in the existing literature that line relaxations and semicoarsening are needed in multigrid algorithms to overcome difficulties caused by the singularities in the axisymmetric Maxwell problems. Our multigrid analysis proceeds by applying the well known regularity based multigrid theory. In order to apply this theory, we prove regularity results for the axisymmetric Laplace and Maxwell equations in certain weighted Sobolev spaces. These, together with some new finite element error estimates in certain weighted Sobolev norms, are the main ingredients of our analysis.

     

Convergence analysis of a multigrid method for convection–diffusion equations

Summary. This paper is concerned with the convergence analysis of robust multigrid methods for convection-diffusion problems. We consider a finite difference discretization of a 2D model convection-diffusion problem with constant coefficients and Dirichlet boundary conditions. For the approximate solution of this discrete problem a multigrid method based on semicoarsening, matrix-dependent prolongation and restriction and line smoothers is applied. For a multigrid W-cycle we prove an upper bound for the contraction number in the euclidean norm which is smaller than one and independent of the mesh size and the diffusion/convection ratio. For the contraction number of a multigrid V-cycle a bound is proved which is uniform for a class of convection-dominated problems. The analysis is based on linear algebra arguments only. Received April 26, 2000 / Published online June 20, 2001     

ON THE CONVERGENCE OF THE NONNESTED V-CYCLE MULTIGRID METHOD FOR NONSYMMETRIC AND INDEFINITE SECOND-ORDER ELLIPTIC PROBLEMS

This paper provides a proof for the uniform convergence rate （independently of the number of mesh levels） for the nonnested V-cycle multigrid method for nonsymmetric and indefinite second-order elliptic problems.