半线性问题的瀑布型多重网格法

本文提出了求解半线性椭圆问题的一类新的瀑布型多重网格法，在网格层数固定的条件下证明了此法的最优阶收敛性。     

板问题多重网格法收敛性的低模估计

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…     

双参数Morley有限元多重网格法

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。     

经济的瀑布型多重网格法(ECMG)

提出一种新的经济的瀑布型多重网格法(ECMG), 和通常的瀑布型多重网格法(CMG)的工作量相比, 新的瀑布型多重网格法在每层上的工作量 都相应的减少, 尤其是粗网格上的工作量将大量的减少. 新格式的误差和通常的 瀑布型多重网格法一样, 都具有最优精度. 最后给出数值算例 来验证所得理论的结果.     

一类新的瀑布型多重网格法

对椭圆问题提出了一类新的瀑布型多重网格法.     

一类非对称椭圆问题瀑布型多重网格法

本将瀑布型多重网格法用于求解非对称椭圆边值问题，数值结果表明算法是有效的。     

抛物问题非协调元多重网格法

抛物问题非协调元多重网格法周叔子,文承标（湖南大学应用数学系）ＮＯＮＣＯＮＦＯＲＭＩＮＧＥＬＥＭＥＮＴＭＵＬＴＩＧＲＩＤＭＥＴＨＯＤＦＯＲＭＲＡＢＯＬＩＣＥＱＵＡＴＩＯＮＳ￥ＺｈｏｕＳｈｕ－ｚｉ;ＷｅｎＣｈｅｎｇ－ｂｉａｏ（ＨｕｎａｎＵｎｉｖｅｒｓｉ...     

一类非线性椭圆问题的瀑布型多重网格法

本对二阶非线性椭圆问题提出一种瀑布型多重网格法，数值实验表明该算法非常有效，当d＝1时，给出了理论结果。     

油气藏两相渗流问题的多重网格法

油气藏数值模拟要解一套高维、非线性、奇异、不定常方程组,工作量很大.本文讨论非线性椭圆-抛物方程的多重网格算法,给出了四重网格计算子程序,并把它用于底水气田计算,与SOR法比较,计算结果完全相同,计算效率显著提高。     

论非线性多重网格法的逼近性质

多重网格法是一种求解椭圆边值问题离散所得的大型线性或非线性方程组的“最优”解法。在有限元离散情形,Hackbusch提出了一种多重网格法的收敛分析方法,即把线性或非线性的多重网格法收敛率的估计问题归结为所谓“光滑性质”与“逼近性质”的研究。在线性情形,若已知有限元解的误差估计,一般容易得到多重网格法的“逼近性质”。但对非线性多重网格法的“逼近性质”在什么条件下成立,尚未见到这方面的工     

Uniform Convergence of Multigrid V-Cycle on Adaptively Refined Finite Element Meshes for Elliptic Problems with Discontinuous Coefficients

The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered. Under the conditions that the coefficient is quasi-monotone up to a constant and the meshes are locally refined by using the newest vertex bisection algorithm, some uniform convergence results are proved for the standard multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on new nodes and their immediate neighbours. The multigrid V-cycle algorithm uses $\mathcal{O}(N)$ operations per iteration and is optimal.     

An Iterative Multigrid Regularization Method for Toeplitz Discrete Ill-Posed Problems

Iterative regularization multigrid methods have been successfully applied to signal/image deblurring problems. When zero-Dirichlet boundary conditions are imposed the deblurring matrix has a Toeplitz structure and it is potentially full. A crucial task of a multilevel strategy is to preserve the Toeplitz structure at the coarse levels which can be exploited to obtain fast computations. The smoother has to be an iterative regularization method. The grid transfer operator should preserve the regularization property of the smoother. This paper improves the iterative multigrid method proposed in [11] introducing a wavelet soft-thresholding denoising post-smoother. Such post-smoother avoids the noise amplification that is the cause of the semi-convergence of iterative regularization methods and reduces ringing effects. The resulting iterative multigrid regularization method stabilizes the iterations so that the imprecise (over) estimate of the stopping iteration does not have a deleterious effect on the computed solution. Numerical examples of signal and image deblurring problems confirm the effectiveness of the proposed method.     

Cascadic Multigrid for Finite Volume Methods for Elliptic Problems

In this paper, some effective cascadic multigrid methods are proposed for solving the large scale symmetric or nonsymmetric algebraic systems arising from the finite volume methods for second order elliptic problems. It is shown that these algorithms are optimal in both accuracy and computational complexity. Numerical experiments are reported to support our theory.     

Multigrid for the Galerkin least squares method in linear elasticity

In SIAM J. Numer. Anal. 28 (1991) 1680-1697, Franca and Stenberg developed several Galerkin least squares methods for the solution of the problem of linear elasticity. That work concerned itself only with the error estimates of the method. It did not address the related problem of finding effective methods for the solution of the associated linear systems. In this work, we prove the convergence of a multigrid method. This multigrid is robust in that the convergence is uniform as the parameter ν goes to 1/2. Computational experiments are included.     

带补偿C^0非协调板元的多重网格法

本文考虑重调和方程的Ｃ０非协调元逼近．通过双线性型ｃｋ（ｕ,ｖ）引入的补偿和将多重网格法应用到Ｃ０非协调板元,给出了更精确的逼近．     

A Multigrid Scheme for Elliptic Constrained Optimal Control Problems

A multigrid scheme for the solution of constrained optimal control problems discretized by finite differences is presented. This scheme is based on a new relaxation procedure that satisfies the given constraints pointwise on the computational grid. In applications, the cases of distributed and boundary control problems with box constraints are considered. The efficient and robust computational performance of the present multigrid scheme allows to investigate bang-bang control problems.AMS Subject Classification: 49J20, 65N06, 65N12, 65N55Supported in part by the SFB 03 “Optimization and Control”     

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.     

Uniformly Convergent Multigrid Methods for Convection–Diffusion Problems without Any Constraint on Coarse Grids

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.     

有限体积法与多重网格法用于提高激波捕捉质量及加速收敛

多重网格技术是一种非常有效的数值计算方法，本文采用多重网格的ＦＡＳ格式进行数值实验，计算加速效果十分明显，同时，结合矢通量分裂用有限体积法，大大提高了主激波的质量。     

Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients

An algebraic Newton-multigrid method is proposed in order to efficiently solve systems of nonlinear reaction-diffusion problems with stochastic coefficients. These problems model the conversion of starch into sugars in growing apples. The stochastic system is first converted into a large coupled system of deterministic equations by applying a stochastic Galerkin finite element discretization. This method leads to high-order accurate stochastic solutions. A stable and high-order time discretization is obtained by applying a fully implicit Runge-Kutta method. After Newton linearization, a point-based algebraic multigrid solution method is applied. In order to decrease the computational cost, alternative multigrid preconditioners are presented. Numerical results demonstrate the convergence properties, robustness and efficiency of the proposed multigrid methods.