TWO-LEVEL MULTISCALE FINITE ELEMENT METHODS FOR THE STEADY NAVIER-STOKES PROBLEM

In this article, on the basis of two-level discretizations and multiscale finite element method, two kinds of finite element algorithms for steady Navier-Stokes problem are presented and discussed. The main technique is first to use a standard finite element discretization on a coarse mesh to approximate low frequencies, then to apply the simple and Newton scheme to linearize discretizations on a fine grid. At this process, multiscale finite element method as a stabilized method deals with the lowest equal-order finite element pairs not satisfying the inf-sup condition. Under the uniqueness condition, error analyses for both algorithms are given. Numerical results are reported to demonstrate the effectiveness of the simple and Newton scheme.     

Multiscale finite element discretizations based on local defect correction for the biharmonic eigenvalue problem of plate buckling

In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295‐324), we establish a three‐scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correction. We first prove a local priori error estimate of finite element approximations, then give the error estimates of multiscale discretization schemes. Theoretical analysis and numerical experiments indicate that our schemes are suitable and efficient for eigenfunctions with local low smoothness.     

A monotone finite element scheme for convection-diffusion equations

A simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection-diffusion problems in any spatial dimension by properly averaging the PDE coefficients on element edges. The resulting finite element stiffness matrix is an -matrix under some mild assumption for the underlying (generally unstructured) finite element grids. As a consequence the proposed edge-averaged finite element scheme is particularly interesting for the discretization of convection dominated problems. This scheme admits a simple variational formulation, it is easy to analyze, and it is also suitable for problems with a relatively smooth flux variable. Some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems.

     

Stabilized Crouzeix–Raviart element for Darcy–Forchheimer model

In this article, a stabilized mixed finite element method for steady Darcy–Forchheimer flow is introduced, in which the velocity and pressure are approximated by nonconforming Crouzeix–Raviart element and piecewise constant, respectively. A discrete inf‐sup condition and a priori error estimates are derived. An iterative scheme is given for practical computation. Finally, some numerical examples are carried out to verify the theoretical analysis and a comparison between two discretizations is given to demonstrate that one of the discretizations has better properties. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1568–1588, 2015     

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 OF THE CRANK-NICOLSON/NEWTON SCHEME FOR NONLINEAR PARABOLIC PROBLEM

In this paper, the Crank-Nicolson/Newton scheme for solving numerically secondorder nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the spatial discretization of the problem and the Crank-Nicolson/Newton scheme is applied to the time discretization of the resulted finite element equations. Moreover, assuming the appropriate regularity of the exact solution and the finite element solution, we obtain optimal error estimates of the fully discrete CrankNicolson/Newton scheme of nonlinear parabolic problem. Finally, numerical experiments are presented to show the efficient performance of the proposed scheme.     

LOCAL AND PARALLEL FINITE ELEMENT ALGORITHMS FOR THE NAVIER-STOKES PROBLEM

Based on two-grid discretizations, in this paper, some new local and parallel finiteelement algorithms are proposed and analyzed for the stationary incompressible Navier-Stokes problem. These algorithms are motivated by the observation that for a solutionto the Navier-Stokes problem, low frequency components can be approximated well by arelatively coarse grid and high frequency components can be computed on a fine grid bysome local and parallel procedure. One major technical tool for the analysis is some locala priori error estimates that are also obtained in this paper for the finite element solutionson general shape-regular grids.     

Local and parallel finite element algorithms based on two-grid discretizations for the transient Stokes equations

Based on two-grid discretizations, some local and parallel finite element algorithms for the d-dimensional (d = 2,3) transient Stokes equations are proposed and analyzed. Both semi- and fully discrete schemes are considered. With backward Euler scheme for the temporal discretization, the basic idea of the fully discrete finite element algorithms is to approximate the generalized Stokes equations using a coarse grid on the entire domain, then correct the resulted residue using a finer grid on overlapped subdomains by some local and parallel procedures at each time step. By the technical tool of local a priori estimate for the fully discrete finite element solution, errors of the corresponding solutions from these algorithms are estimated. Some numerical results are also given which show that the algorithms are highly efficient.     

Mesh-Independence of the Lagrange–Newton Method for Nonlinear Optimal Control Problems and their Discretizations

In a recent paper we proved a mesh-independence principle for Newton's method applied to stable and consistent discretizations of generalized equations. In this paper we introduce a new consistency condition which is easier to check in applications. Using this new condition we show that the mesh-independence principle holds for the Lagrange–Newton method applied to nonlinear optimal control problems with mixed control-state constraints and their discretizations by Euler's method or Ritz type methods.     

求解两类Maxwell方程组棱元离散系统的快速算法和自适应方法

Maxwell方程组棱元离散系统的快速算法和自适应方法是当前计算电磁场中的研究热点和难点. 首先, 针对H(curl)椭圆方程组的棱元离散系统, 通过建立棱元空间的稳定性分解, 设计了相应的快速迭代法和高效预条件子, 并且证明了迭代算法的收敛率和预条件子的条件数均不依赖于模型参数和网格规模. 其次, 针对时谐Maxwell方程组的棱有限元方法, 利用离散的Helmholtz分解, 连续散度为零函数对离散散度为零函数的逼近性和对偶论证, 获得了在L²和H(curl)范数下的拟最优误差估计. 进而设计和分析了相应的两网格法. 最后, 分别针对变系数H(curl)椭圆方程组和不定时谐Maxwell方程组, 考虑了一种不需要标记振荡项和加密单元不需要满足“内节点” 性质的自适应棱有限元法(AEFEM), 并证明了AEFEM的收敛性. 进一步, 当初始网格和Dörfler标记策略参数满足一定的假设条件时, 利用AEFEM的收敛性、误差的整体下界和局部上界估计, 证明了AEFEM的拟最优复杂性.     

矩阵形式二次修正Maxwell-Dirac系统的多尺度算法

带二次修正项的Dirac方程在拓扑绝缘体、石墨烯、超导等新材料电磁光特性分析中有着十分广泛的应用.本文工作的创新点有：一是首次提出了矩阵形式带有二次修正项的Dirac方程,它是比较一般的数学框架,涵盖了上述材料体系很多重要的物理模型,具体见附录A;二是针对上述材料体系的电磁响应问题,提出了有界区域Weyl规范下具有周期间断系数矩阵形式带二次修正项Maxwell-Dirac系统的多尺度渐近方法,结合Crank-Nicolson有限差分方法和自适应棱单元方法,发展了一类多尺度算法.数值试验结果验证了多尺度渐近方法的正确性和算法的有效性.     

Local and Parallel Finite Element Algorithms Based on Two-Grid Discretizations for Nonlinear Problems

In this paper, some local and parallel discretizations and adaptive finite element algorithms are proposed and analyzed for nonlinear elliptic boundary value problems in both two and three dimensions. The main technique is to use a standard finite element discretization on a coarse grid to approximate low frequencies and then to apply some linearized discretization on a fine grid to correct the resulted residual (which contains mostly high frequencies) by some local/parallel procedures. The theoretical tools for analyzing these methods are some local a priori and a posteriori error estimates for finite element solutions on general shape-regular grids that are also obtained in this paper.     

Multilevel preconditioners for discretizations of the biharmonic equation by rectangular finite elements

Recently, some new multilevel preconditioners for solving elliptic finite element discretizations by iterative methods have been proposed. They are based on appropriate splittings of the finite element spaces under consideration, and may be analyzed within the framework of additive Schwarz schemes. In this paper we discuss some multilevel methods for discretizations of the fourth-order biharmonic problem by rectangular elements and derive optimal estimates for the condition numbers of the preconditioned linear systems. For the Bogner–Fox–Schmit rectangle, the generalization of the Bramble–Pasciak–Xu method is discussed. As a byproduct, an optimal multilevel preconditioner for nonconforming discretizations by Adini elements is also derived.     

Two‐grid mixed finite‐element methods for nonlinear Schrödinger equations

Two‐grid mixed finite element schemes are developed for solving both steady state and unsteady state nonlinear Schrödinger equations. The schemes use discretizations based on a mixed finite‐element method. The two‐grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all of the Newton‐like iterations to grids much coarser than the final one, with no loss in order of accuracy. Numerical tests are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 63‐73, 2012     

Multiscale methods for elliptic homogenization problems

In this article we study two families of multiscale methods for numerically solving elliptic homogenization problems. The recently developed multiscale finite element method [Hou and Wu, J Comp Phys 134 (1997), 169–189] captures the effect of microscales on macroscales through modification of finite element basis functions. Here we reformulate this method that captures the same effect through modification of bilinear forms in the finite element formulation. This new formulation is a general approach that can handle a large variety of differential problems and numerical methods. It can be easily extended to nonlinear problems and mixed finite element methods, for example. The latter extension is carried out in this article. The recently introduced heterogeneous multiscale method [Engquist and Engquist, Comm Math Sci 1 (2003), 87–132] is designed for efficient numerical solution of problems with multiscales and multiphysics. In the second part of this article, we study this method in mixed form (we call it the mixed heterogeneous multiscale method). We present a detailed analysis for stability and convergence of this new method. Estimates are obtained for the error between the homogenized and numerical multiscale solutions. Strategies for retrieving the microstructural information from the numerical solution are provided and analyzed. Relationship between the multiscale finite element and heterogeneous multiscale methods is discussed. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

TWO-SCALE FINITE ELEMENT DISCRETIZATIONS FOR PARTIAL DIFFERENTIAL EQUATIONS

Some two-scale finite element discretizations are introduced for a class of linear partialdifferential equations. Both boundary value and eigenvalue problems are studied. Basedon the two-scale error resolution techniques, several two-scale finite element algorithmsare proposed and analyzed. It is shown that this type of two-scale algorithms not onlysignificantly reduces the number of degrees of freedom but also produces very accurateapproximations.     

分数阶Burgers方程的有限元计算(英文)

建立了一维和二维分数阶Burgers方程的有限元格式.时间分数阶导数使用L1方法离散,空间方向使用有限元方法离散.通过选择合适的基函数,将离散后的方程转化成一个非线性代数方程组,并应用牛顿迭代方法求解.数值实验显示出了方法的有效性.     

Two-level defect-correction locally stabilized finite element method for the steady Navier–Stokes equations

This paper proposes a two-level defect-correction stabilized finite element method for the steady Navier–Stokes equations based on local Gauss integration. The method combines the two-level strategy with the defect-correction method under the assumption of the uniqueness condition. Both the simplified and the Newton scheme are proposed and analyzed. Moreover, the numerical illustrations agree completely with the theoretical expectations.     

Radial basis collocation method and quasi‐Newton iteration for nonlinear elliptic problems

This work presents a radial basis collocation method combined with the quasi‐Newton iteration method for solving semilinear elliptic partial differential equations. The main result in this study is that there exists an exponential convergence rate in the radial basis collocation discretization and a superlinear convergence rate in the quasi‐Newton iteration of the nonlinear partial differential equations. In this work, the numerical error associated with the employed quadrature rule is considered. It is shown that the errors in Sobolev norms for linear elliptic partial differential equations using radial basis collocation method are bounded by the truncation error of the RBF. The combined errors due to radial basis approximation, quadrature rules, and quasi‐Newton and Newton iterations are also presented. This result can be extended to finite element or finite difference method combined with any iteration methods discussed in this work. The numerical example demonstrates a good agreement between numerical results and analytical predictions. The numerical results also show that although the convergence rate of order 1.62 of the quasi‐Newton iteration scheme is slightly slower than rate of order 2 in the Newton iteration scheme, the former is more stable and less sensitive to the initial guess. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008