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1.
The multigrid waveform relaxation (WR) algorithm has been fairly studied and implemented for parabolic equations. It has been found that the performance of the multigrid WR method for a parabolic equation is practically the same as that of multigrid iteration for the associated steady state elliptic equation. However, the properties of the multigrid WR method for hyperbolic problems are relatively unknown. This paper studies the multigrid acceleration to the WR iteration for hyperbolic problems, with a focus on the convergence comparison between the multigrid WR iteration and the multigrid iteration for the corresponding steady state equations. Using a Fourier-Laplace analysis in two case studies, it is found that the multigrid performance on hyperbolic problems no longer shares the close resemblance in convergence factors between the WR iteration for parabolic equations and the iteration for the associated steady state equations. 相似文献
2.
Jari Toivanen & Cornelis W. Oosterlee 《高等学校计算数学学报(英文版)》2012,5(1):85-98
We present an algebraic version of an iterative multigrid method for
obstacle problems, called projected algebraic multigrid (PAMG) here.
We show that classical algebraic multigrid algorithms can easily be
extended to deal with this kind of problem. This paves the way for
efficient multigrid solution of obstacle problems with partial
differential equations arising, for example, in financial engineering. 相似文献
3.
Cascadic multigrid technique for mortar Wilson finite element method of homogeneous boundary value planar linear elasticity is described and analyzed. First the mortar Wilson finite element method for planar linear elasticity will be analyzed, and the error estimate under L2 and H1 norm is optimal. Then a cascadic multigrid method for the mortar finite element discrete problem is described. Suitable grid transfer operator and smoother are developed which lead to an optimal cascadic multigrid method. Finally, the computational results are presented. 相似文献
4.
1.引言 近年来,多重网格法已成为行之有效的偏微分方程数值解法.对板问题有限元离散系统的多重网格法,也有不少的研究工作,如[4],[5],[10],[13-17].在[4],[14-17]中,作者讨论了C1协调元离散板问题的多重网格法,并在能量模(即 H2模)意义下获得了最优的收敛率.在[5],[10]中,作者讨论了非协调元离散问题的多重网格法,并在能量模意义下获得了最优的收敛率,同时在能量模意义下证明了套迭代多重网格法一阶收敛.但对板问题多重网格法的低模估计,即 H1模估计,至今尚未见研究,本文… 相似文献
5.
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. 相似文献
6.
In this paper, we design and analyze an algebraic multigrid method for a condensed finite element system on criss-cross grids
and then provide a convergence analysis. Criss-cross grid finite element systems represent a large class of finite element
systems that can be reduced to a smaller system by first eliminating certain degrees of freedoms. The algebraic multigrid
method that we construct is analogous to many other algebraic multigrid methods for more complicated problems such as unstructured
grids, but, because of the specialty of our problem, we are able to provide a rigorous convergence analysis to our algebraic
multigrid method.
Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday
The work was supported in part by NSAF(10376031) and National Major Key Project for basic researches and by National High-Tech
ICF Committee in China. 相似文献
7.
Akira IMAKURA 《数学研究及应用》2021,41(1):87-98
Multigrid methods are widely used and well studied for linear solvers and preconditioners of Krylov subspace methods. The multigrid method is one of the most powerful approaches for solving large scale linear systems;however, it may show low parallel efficiency on coarse grids. There are several kinds of research on this issue. In this paper, we intend to overcome this difficulty by proposing a novel multigrid algorithm that has multiple grids on each layer.Numerical results indicate that the proposed method shows a better convergence rate compared with the existing multigrid method. 相似文献
8.
A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term. 相似文献
9.
Qian-shunChang Wei-weiSun 《计算数学(英文版)》2005,23(2):177-184
In this paper, we consider multigrid methods for solving symmetric nonnegative definite matrix equations. We present some interesting features of the multigrid method and prove that the method is convergent in L2 space and the convergent solution is unique for such nonnegative definite system and given initial guess. 相似文献
10.
Barbara Kaltenbacher. 《Mathematics of Computation》2003,72(244):1711-1730
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.
11.
The multigrid method is compared to ICCG/MICCG methods for solvingsymmetric systems of linear equations arising from approximationsto differential equations with jump discontinuities in the coefficients.An optimal multigrid algorithm for these types of problems isdeveloped. It includes pattern relaxation and acceleration.Optimization of ICCG/MICCG algorithms is investigated. Thisincludes the effect of adding extra (up to ten) bands to theapproximate factorization and of different grid ordering schemes.Numerical results are presented comparing the scalar work ofthe algorithms. For large problems the multigrid algorithm issuperior. The optimal multigrid scheme can be highly vectorized. 相似文献
12.
13.
In this paper, multigrid methods with residual scaling techniques for symmetric positive definite linear systems are considered. The idea of perturbed two-grid methods proposed in [7] is used to estimate the convergence factor of multigrid methods with residual scaled by positive constant scaling factors. We will show that if the convergence factors of the two-grid methods are uniformly bounded by σ (σ<0.5), then the convergence factors of the W-cycle multigrid methods are uniformly bounded by σ/(1−σ), whether the residuals are scaled at some or all levels. This result extends Notay’s Theorem 3.1 in [7] to more general cases. The result also confirms the viewpoint that the W-cycle multigrid method will converge sufficiently well as long as the convergence factor of the two-grid method is small enough. In the case where the convergence factor of the two-grid method is not small enough, by appropriate choice of the cycle index γ, we can guarantee that the convergence factor of the multigrid methods with residual scaling techniques still has a uniform bound less than σ/(1−σ). Numerical experiments are provided to show that the performance of multigrid methods can be improved by scaling the residual with a constant factor. The convergence rates of the two-grid methods and the multigrid methods show that the W-cycle multigrid methods perform better if the convergence rate of the two-grid method becomes smaller. These numerical experiments support the proposed theoretical results in this paper. 相似文献
14.
Hosae Lee 《Journal of Applied Mathematics and Computing》1997,4(2):427-440
In this paper, we introduce a multigrid method for solving the nonliear Urysohn integral equation. The algorithm is derived from a discrete resolvent equation which approximates the continuous resolvent equation of the nonlinear Urysohn integral equation. The algorithm is mathematically equivalent to Atkinson’s adaptive twogrid iteration. But the two are different computationally. We show the convergence of the algorithm and its equivalence to Atkinson’s adaptive twogrid iteration. In our numerical example, we compare our algorithm to other multigrid methods for solving the nonliear Urysohn integral equation including the nonlinear multigrid method introduced by Hackbush. 相似文献
15.
Hermann Weber 《Numerische Mathematik》1993,66(1):525-541
Summary We introduce a multigrid method for the solution of the discrete Stokes equations, arising from a Petrov-Galerkin formulation. The stiffness matrix is nonsymmetric but coercive, hence we consider smoothing iterations which are not suitable for usual indefinite problems. In this report, we prove convergence for a multigrid method with Richardson iteration in the smoothing part. 相似文献
16.
Sheng Zhang Dehao Yu 《计算数学(英文版)》2007,25(1):13-26
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. 相似文献
17.
The cascadic multigrid method for elliptic problems 总被引:23,自引:0,他引:23
Summary. The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods
based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade
principle view, a basic iteration method without any preconditioner is used at successive refinement levels. For a prescribed
error tolerance on the final level, more iterations must be spent on coarser grids in order to allow for less iterations on
finer grids. A first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of [9], in short CCG method, whichused the CG method as basic iteration method on each level. In [18] it has been proven,
that the CCG method is accurate with optimal complexity for elliptic problems in 2D and quasi-uniform triangulations. The
present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid
smoothers. Moreover, an adaptive control strategy for the number of iterations on successive refinement levels for possibly
highly non-uniform grids is worked out on the basis of a posteriori estimates. Numerical tests confirm the efficiency and
robustness of the cascadic multigrid method.
Received November 12, 1994 / Revised version received October 12, 1995 相似文献
18.
Chun-jiaBi Li-kangLi 《计算数学(英文版)》2004,22(1):123-136
The purpose of this paper is to study the cascadic multigrid method for the secondorder elliptic problems with curved boundary in two-dimension which are discretized by the isoparametric finite element method with numerical integration. We show that the CCG method is accurate with optimal complexity and traditional multigrid smoother (likesymmetric Gauss-Seidel, SSOR or damped Jacobi iteration) is accurate with suboptimal complexity. 相似文献
19.
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. 相似文献
20.
1、引言 多重网格方法是求解偏微分方程的高效快速算法,在实际中得到广泛应用.[2][6]中考察了Morley元的多重网格方法,并用于双调和方程问题。 相似文献