拟等级网格下非线性延迟微分方程间断有限元法

本文利用间断有限元法求解非线性延迟微分方程,在拟等级网格下.给出非线性延迟微分方程间断有限元解的整体收敛阶和局部超收敛阶,数值实验验证了理论结果的正确性.     

曲边区域上的多边形网格间断有限元离散及其多重网格算法

本文利用多边形网格上的间断有限元方法离散二阶椭圆方程,在曲边区域上,采用多条直短边逼近曲边的以直代曲的策略,实现了高阶元在能量范数下的最优收敛.本文还将这一方法用于带曲边界面问题的求解,同样得到高阶元的最优收敛.此外我们还设计并分析了这一方法的\linebreakW-cycle和Variable V-cycle多重网格预条件方法,证明当光滑次数足够多时,多重网格预条件算法一致收敛.最后给出了数值算例,证实该算法的可行性并验证了理论分析的结果.     

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.     

用Fourier Galerkin方法求解间断特征值问题

本文我们考虑一个带间断系数的特征值问题,使用Fourier G a lerk in方法求解.数值试验表明特征值收敛速度达到三阶,而特征函数收敛速度达到2.5阶.表明此方法对间断系数问题非常有效.     

并行间断有限元算法求解Navier-Stokes方程

间断Galerkin有限元方法非常适合在非结构网格上高精度求解Navier-Stokes方程,然而其十分耗费计算资源.为了提高计算效率,提出了高效的MIMD并行算法.采用隐式时间离散GMRES+LU SGS格式,结合多重网格方法,当地时间步长加速算法收敛.为了保证各处理器间负载平衡,采用区域分解二级图方法划分网格,实现内存合理分配,数据只在相邻处理器间传递.数值模拟了RAE2822翼型和M6黏性绕流,加速比基本呈线性变化且接近理想值.结果表明了该算法能有效减少计算时间、合理分配内存,具有较高的加速比和并行效率,适合于MIMD粗粒度科学计算.     

中子输运方程误差估计及自适应计算

 本文研究了全离散方法求解二维中子输运方程的有限元自适应算法, 角度变量用离散纵坐标方法展开, 空间变量用间断元方法求解. 基于间断元方法给出了空间离散的残量型后验误差估计. 在后验误差估计的基础上, 我们设计了自适应有限元算法.由残量型后验估计可以给出局部加密网格的自适应算法. 最后, 我们给出了数值算例来验证我们的理论结果.     

一种守恒型间断跟踪法中对任意多个间断的移动和相互作用的处理

对一种守恒型间断跟踪法设计了一种技巧来处理任意多个间断的移动和相互作用．由此技巧我们就可以建立一个“一般的强健的”间断跟踪法．由于采用了此技巧就会使得算法在某时刻在某网格上会存在非相容性并且会产生O(1)-强度的误差．但不管怎样,这些误差在后续的计算中会被算法的守恒性所消除．还给出了几个数值例子来显示这一技巧的有效性．     

对流扩散方程的时空间断有限元方法

研究对流扩散方程的时空间断Galerkin有限元方法,该方法采用时,空两个变量都允许间断的基函数,更适用于移动网格,自适应算法以及并行计算.本文利用拉格朗日欧拉方法,采用F.Brezzi数值流通量,给出对流扩散方程的间断时空有限元离散格式,并证明格式的相容性,强制性,稳定性,解的存在唯一性,以及总体误差估计.     

用LDG方法求解奇异摄动Volterra积分微分方程

本文介绍局部间断有限元(LDG)方法,构造求解奇异摄动Volterra积分微分方程的LDG数值格式.算例表明,在局部加密网格下,LDG解的数值通量在节点处具有2p+1阶的一致超收敛性质.     

关于抛物型方程离散流量的收敛性

关于用差分方法求解具有间断系数的二阶抛型方程的问题, Ａ．Ｈ． ＴＮＸＯＨＯＢ与 Ａ．Ａ．Ｃａｍａｐｃ　从１９６１年山开始曾经作过详尽的研究,他们的结果都总结在专著［２］中,有关的文献也可以在该书中找到．他们指出在系数间断点处附近的网格点上格式的截断误差为Ｏ（１）,但差分格式的解在极大意义下收敛于原微分方程的连续解．他们在构造差分格式,并论证其收敛性时,充分利用了原微分方程中流连续的性质,但是却没有讨论差分格式中的离散流量的收敛性．８０年代 Ｔ．Ａ． Ｍａｎｔｅｎｆｆｅｌ, Ａ．Ｂ． Ｗｈｉｔｅ, Ｊｒ…     

A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients

In this paper, we propose a multilevel preconditioner for the Crouzeix-Raviart finite element approximation of second-order elliptic partial differential equations with discontinuous coefficients. Since the finite element spaces are nonnested, weighted intergrid transfer operators, which are stable under the weighted L2 norm, are introduced to exchange information between different meshes. By analyzing the eigenvalue distribution of the preconditioned system, we prove that except a few small eigenvalues, all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and the mesh size. As a result, we get that the convergence rate of the preconditioned conjugate gradient method is quasi-uniform with respect to the jump and the mesh size. Numerical experiments are presented to confirm our theoretical analysis.     

Stabilization of algebraic multilevel iteration methods; additive methods

There exist two main versions of preconditioners of algebraic multilevel type, the additive and the multiplicative methods. They correspond to preconditioners in block diagonal and block matrix factorized form, respectively. Both can be defined and analysed as recursive two-by-two block methods. Although the analytical framework for such methods is simple, for many finite element approximations it still permits the derivation of the strongest results, such as optimal, or nearly optimal, rate of convergence and optimal, or nearly optimal order of computational complexity, when proper recursive global orderings of node points have been used or when they are applied for hierarchical basis function finite element methods for elliptic self-adjoint equations and stabilized in a certain way. This holds for general elliptic problems of second order, independent of the regularity of the problem, including independence of discontinuities of coefficients between elements and of anisotropy. Important ingredients in the methods are a proper balance of the size of the coarse mesh to the finest mesh and a proper solver on the coarse mesh. This paper presents in a survey form the basic results of such methods and considers in particular additive methods. This method has excellent parallelization properties. This revised version was published online in June 2006 with corrections to the Cover Date.     

A coupled continuous-discontinuous FEM approach for convection diffusion equations

In this article, we introduce a coupled approach of local discontinuous Galerkin and standard finite element method for solving convection diffusion problems. The whole domain is divided into two disjoint subdomains. The discontinuous Galerkin method is adopted in the subdomain where the solution varies rapidly, while the standard finite element method is used in the other subdomain due to its lower computational cost. The stability and a priori error estimate are established. We prove that the coupled method has O((ε1 / 2 + h 1 / 2 )h k ) convergence rate in an associated norm, where ε is the diffusion coefficient, h is the mesh size and k is the degree of polynomial. The numerical results verify our theoretical results. Moreover, 2k-order superconvergence of the numerical traces at the nodes, and the optimal convergence of the errors under L 2 norm are observed numerically on the uniform mesh. The numerical results also indicate that the coupled method has the same convergence order and almost the same errors as the purely LDG method.     

BDDC methods for discontinuous Galerkin discretization of elliptic problems

A discontinuous Galerkin (DG) discretization of Dirichlet problem for second-order elliptic equations with discontinuous coefficients in 2-D is considered. For this discretization, balancing domain decomposition with constraints (BDDC) algorithms are designed and analyzed as an additive Schwarz method (ASM). The coarse and local problems are defined using special partitions of unity and edge constraints. Under certain assumptions on the coefficients and the mesh sizes across ∂Ω_i, where the Ω_i are disjoint subregions of the original region Ω, a condition number estimate C(1+max_ilog(H_i/h_i))² is established with C independent of h_i, H_i and the jumps of the coefficients. The algorithms are well suited for parallel computations and can be straightforwardly extended to the 3-D problems. Results of numerical tests are included which confirm the theoretical results and the necessity of the imposed assumptions.     

New convergence estimates for multilevel algorithms for finite-element approximations

New convergence estimates are established for some multilevel algorithms for finite-element methods applied to elliptic problems with jump coefficients. A uniform rate of convergence is derived if the coefficient has only one jump interface. If the coefficient has multi-jump interfaces which meet at only one interior point in the domain, the convergence rate is bounded by 1−(CJ)⁻¹, where J is the number of levels and C is a constant independent of the jump.     

Robust iterative methods for elliptic problems with highly varying coefficients in thin substructures

Summary. In this paper we introduce a class of robust multilevel interface solvers for two-dimensional finite element discrete elliptic problems with highly varying coefficients corresponding to geometric decompositions by a tensor product of strongly non-uniform meshes. The global iterations convergence rate is shown to be of the order with respect to the number of degrees of freedom on the single subdomain boundaries, uniformly upon the coarse and fine mesh sizes, jumps in the coefficients and aspect ratios of substructures. As the first approach, we adapt the frequency filtering techniques [28] to construct robust smoothers on the highly non-uniform coarse grid. As an alternative, a multilevel averaging procedure for successive coarse grid correction is proposed and analyzed. The resultant multilevel coarse grid preconditioner is shown to have (in a two level case) the condition number independent of the coarse mesh grading and jumps in the coefficients related to the coarsest refinement level. The proposed technique exhibited high serial and parallel performance in the skin diffusion processes modelling [20] where the high dimensional coarse mesh problem inherits a strong geometrical and coefficients anisotropy. The approach may be also applied to magnetostatics problems as well as in some composite materials simulation. Received December 27, 1994     

Convergence analysis of the LDG method applied to singularly perturbed problems

Considering a two‐dimensional singularly perturbed convection–diffusion problem with exponential boundary layers, we analyze the local discontinuous Galerkin (DG) method that uses piecewise bilinear polynomials on Shishkin mesh. A convergence rate O(N^‐1 lnN) in a DG‐norm is established under the regularity assumptions, while the total number of mesh points is O(N²). The rate of convergence is uniformly valid with respect to the singular perturbation parameter ε. Numerical experiments indicate that the theoretical error estimate is sharp. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Multilevel Schwarz methods

Summary We consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations. A class of multilevel algorithms are studied using the additive Schwarz framework. We establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels. This is an improvement on Dryja and Widlund's result on a multilevel additive Schwarz algorithm, as well as Bramble, Pasciak and Xu's result on the BPX algorithm. Some multiplicative variants of the multilevel methods are also considered. We establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one, which is independent of the number of levels. For a proper ordering, the iteration operators correspond to the error propagation operators of certain V-cycle multigrid methods, using Gauss-Seidel and damped Jacobi methods as smoothers, respectively.This work was supported in part by the National Science Foundation under Grants NSF-CCR-8903003 at Courant Institute of Mathematical Sciences, New York University and NSF-ASC-8958544 at Department of Computer Science, University of Maryland.     

Rate of Convergence for some constraint decomposition methods for nonlinear variational inequalities

Summary. Some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset. One of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods. For domain decomposition and multigrid methods, the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems. In order to analyse the convergence rate, we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints. A special nonlinear interpolation operator is introduced for decomposing the functions. Received December 13, 2001 / Revised version received February 19, 2002 / Published online June 17, 2002 This work was partially supported by the Norwegian Research Council under projects 128224/431 and SEP-115837/431.     

Local Multilevel Method on Adaptively Refined Meshes for Elliptic Problems with Smooth Complex Coefficients

In this paper, a local multilevel algorithm is investigated for solving linear systems arising from adaptive finite element approximations of second order elliptic problems with smooth complex coefficients. It is shown that the abstract theory for local multilevel algorithm can also be applied to elliptic problems whose dominant coefficient is complex valued. Assuming that the coarsest mesh size is sufficiently small, we prove that this algorithm with Gauss-Seidel smoother is convergent and optimal on the adaptively refined meshes generated by the newest vertex bisection algorithm. Numerical experiments are reported to confirm the theoretical analysis.