A Two-Level Method for Nonsymmetric Eigenvalue Problems

A two-level discretization method for eigenvalue problems is studied. Compared to the standard Galerkin finite element discretization technique performed on a fine grid this method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problems for the case of eigenvalue approximation of nonsymmetric problems). The improved solution has the asymptotic accuracy of the Galerkin discretization solution. The link between the method and the iterated Galerkin method is established. Error estimates for the general nonsymmetric case are derived.     

OPTIMALLY ACCURATE PETROV-GALERKIN METHOD OF FINITE ELEMENTS

We perform analysis for a finite elements method applied to the singular self-adjoint problem.This method uses continuous piecewise polynomial spaces for the trial and the test spaces.We fit the trial polynomial space by piecewise exponentials and we apply so exponentially fitted Galerkin method to singular self-adjomt problem by approximating driving terms by Lagrange piecewise polynomials,linear,quadratic and cubic.Wt measure the erroe in max norm.We show that method is optimal of the first order in the error estimate,We also give numerical results for the Galerkin approximation.     

ACTA MATHEMATICAE APPLICATAE SINICA(English Series)

2005年第21卷第1期摘要A TWO-LEVEL METHOD FOR NONSYMMETRIC EIGENVALUE PROBLEMS -Karel Kolman A two-level discretization method for eigenvalue problems is studied. Compared to the standard Galerkin finite element discretization technique performed on a fine grid this method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problems for the case of eigenvalue approximation of nonsymmetric problems). The improved solution has the asymptotic accuracy of the Galerkin discretization solution. The link between the method and the iterated Galerkin method is established. Error estimates for the general nonsymmetric case are derived.     

耗散型发展方程的一步Newton法

By taking example of the unsteady Navier-Stokes equation, a kind of postpro-cessing method for the standard Galerkin approximation, which is called one step Newton method for simplicity, is proposed by applying the idea of Newton itera-tion to unsteady problems. The analysis results show that this method can greatly improve the accuracy of the standard Galerkin approximation and the numerical experiments also indicate that it is a high performance method.     

A MIXED FINITE ELEMENT METHOD FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE

The transient behavior of a semiconductor device is described by a system of three quasilinear partial differential equations. One is elliptic in form for the electric potential and the other two are parabolic in form for the conservation of electron and hole concentrations. The electric potential equation is discretized by a mixed finite element method. The electron and hole density equations are treated by a Galerkin method that applies a variant of the method of characteristics to the transport terms. Optimal order convergence analysis in L2 is given for the proposed method.     

Superconvergence of a combined mixed finite element and discontinuous Galerkin method for a compressible miscible displacement problem

A combined mixed finite element and discontinuous Galerkin method for a compressible miscible displacement problem which includes molecular diffusion and dispersion in porous media is investigated. That is to say, the mixed finite element method with Raviart-Thomas space is applied to the flow equation, and the transport one is solved by the symmetric interior penalty discontinuous Galerkin (SIPG) approximation. Based on projection interpolations and induction hypotheses, a superconvergence estimate is obtained. During the analysis, an extension of the Darcy velocity along the Gauss line is also used in the evaluation of the coefficients in the Galerkin procedure for the concentration.     

EXISTENCE AND REGULARITY OF SOLUTIONS TO MODEL FOR LIQUID MIXTURE OF ^3HE-^4HE

Existence and regularity of solutions to model for liquid mixture of 3He-4He is considered in this paper.First,it is proved that this system possesses a unique global weak solution in H 1(,C × R) by using Galerkin method.Secondly,by using an iteration procedure,regularity estimates for the linear semigroups,it is proved that the model for liquid mixture of 3He-4He has a unique solution in Hk(,C × R) for all k ≥ 1.     

A NEW DIRECT DISCONTINUOUS GALERKIN METHOD WITH SYMMETRIC STRUCTURE FOR NONLINEAR DIFFUSION EQUATIONS

In this paper we continue the study of discontinuous Galerkin finite element methods for nonlinear diffusion equations following the direct discontinuous Galerkin （DDG） meth- ods for diffusion problems [17] and the direct discontinuous Galerkin （DDG） methods for diffusion with interface corrections [18]. We introduce a numerical flux for the test func- tion, and obtain a new direct discontinuous Galerkin method with symmetric structure. Second order derivative jump terms are included in the numerical flux formula and explicit guidelines for choosing the numerical flux are given. The constructed scheme has a sym- metric property and an optimal L2 （L2） error estimate is obtained. Numerical examples are carried out to demonstrate the optimal （k ＋ 1）th order of accuracy for the method with pk polynomial approximations for both linear and nonlinear problems, under one-dimensional and two-dimensional settings.     

APPLICATION OF SUPERCONVERGENCE TO A MODEL FOR COMPRESSIBLE MISCIBLE DISPLACEMENT

A modification of a finite element method of Douglas and Roberts for approximating the solution of the equations describing compressible miscible displacement in a porous medium is proposed and analyzed. The pressure is treated by a parabolic mixed finite element method using a Raviart-Thomas space of index rover a quasiregular partition, An extension of the Darcy velocity along Gauss lines is used in the evaluation of the coefficients in the Galerkin procedure for the concentration. A simple computational procedure allows the superconvergence property of the fluid velocity to be retained in our total algorithm.     

SUPER-GEOMETRIC CONVERGENCE OF A SPECTRAL ELEMENT METHOD FOR EIGENVALUE PROBLEMS WITH JUMP COEFFICIENTS

We propose and analyze a C^0 spectral element method for a model eigenvalue problem with discontinuous coefficients in the one dimensional setting. A super-geometric rate of convergence is proved for the piecewise constant coefficients case and verified by numerical tests. Furthermore, the asymptotical equivalence between a Gauss-Lobatto collocation method and a spectral Galerkin method is established for a simplified model.     

Singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel

In a recent paper [3], Cao and Xu established the Galerkin method for weakly singular Fredholm integral equations that preserves the singularity of the solution. Their Galerkin method provides a numerical solution that is a linear combination of a certain class of basis functions which includes elements that reflect the singularity of the solution. The purpose of this paper is to extend the result of Cao and Xu and to establish the singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. The iterated singularity preserving Galerkin method is also discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.     

Stability and convergence of optimum spectral non‐linear Galerkin methods

Our objective in this article is to present some numerical schemes for the approximation of the 2‐D Navier–Stokes equations with periodic boundary conditions, and to study the stability and convergence of the schemes. Spatial discretization can be performed by either the spectral Galerkin method or the optimum spectral non‐linear Galerkin method; time discretization is done by the Euler scheme and a two‐step scheme. Our results show that under the same convergence rate the optimum spectral non‐linear Galerkin method is superior to the usual Galerkin methods. Finally, numerical example is provided and supports our results. Copyright © 2001 John Wiley & Sons, Ltd.     

ON THE BREAKDOWNS OF THE GALERKIN AND LEAST-SQUARES METHODS

1　ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｃｏｎｓｉｄｅｒｌｉｎｅａｒｓｙｓｔｅｍｓｏｆｔｈｅｆｏｒｍＡｘ=ｂ,(1 )ｗｈｅｒｅＡ∈ＣＮ×Ｎｉｓｎｏｎｓｉｎｇｕｌａｒａｎｄｐｏｓｓｉｂｌｙｎｏｎ Ｈｅｒｍｉｔｉａｎ .Ａｍａｊｏｒｃｌａｓｓｏｆｍｅｔｈｏｄｓｆｏｒｓｏｌｖｉｎｇ (1 )ｉｓｔｈｅｃｌａｓｓｏｆＫｒｙｌｏｖｓｕｂｓｐａｃｅｍｅｔｈｏｄｓ (ｓｅｅ[6] ,[1 3]ｆｏｒｏｖｅｒｖｉｅｗｓｏｆｓｕｃｈｍｅｔｈｏｄｓ) ,ｄｅｆｉｎｅｄｂｙｔｈｅｐｒｏｐｅｒｔｉｅｓｘｍ ∈ｘ0 +Ｋｍ(ｒ0 ,Ａ) ;(2 )ｒｍ ⊥Ｌｍ, (3)ｗｈｅ…     

Comparison of the finite difference and Galerkin methods as applied to the solution of the hydrodynamic equations

The three-dimensional nonlinear hydrodynamic equations which describe wind induced flow in a homogeneous sea are transformed from Cartesian coordinates into sigma coordinates. The solution of these equations in the horizontal is accomplished using a standard finite difference grid and established finite difference methods.The accuracy and computational efficiency, in terms of both computer time and main memory requirements, of using either the Galerkin method or a finite difference grid through the vertical is considered. Calculations, using the same number of functions in the Galerkin method as grid bases through the vertical shows that the Galerkin method has superior accuracy over the grid box method. Hence, for a given accuracy a smaller number of functions than grid boxes may be used, with associated saving in computational resources.For the case in which the vertical variation of eddy viscosity is fixed, an eigenvalue problem can be solved to yield a set of eigenfunctions. Using these eigenfunctions as a basis set with the Galerkin approach, a Galerkin-eigenfunction method is developed. Calculations show that the Galerkin-eigenfunction technique is accurate and in a linear model is clearly computationally more economic than the use of grid boxes through the vertical.     

A TWO-LEVEL FINITE ELEMENT GALERKIN METHOD FOR THE NONSTATIONARY NAVIER-STOKES EQUATIONS II： TIME DISCRETIZATION

In this article we consider the fully discrete two-level finite element Galerkin method for the two-dimensional nonstationary incompressible Navier-Stokes equations. This method consists in dealing with the fully discrete nonlinear Navier-Stokes problem on a coarse mesh with width $H$ and the fully discrete linear generalized Stokes problem on a fine mesh with width $h << H$. Our results show that if we choose $H=O(h^{1/2}$) this method is as the same stability and convergence as the fully discrete standard finite element Galerkin method which needs dealing with the fully discrete nonlinear Navier-Stokes problem on a fine mesh with width $h$. However, our method is cheaper than the standard fully discrete finite element Galerkin method.     

轴向变速运动粘弹性弦线横向振动的复模态Galerkin方法

在考虑初始张力和轴向速度简谐涨落的情况下,利用含预应力三维变形体的运动方程,建立了轴向变速运动弦线横向振动的非线性控制方程,材料的粘弹性行为由Kelvin模型描述．利用匀速运动线性弦线的模态函数构造了变速运动非线性弦线复模态Galerkin方法的基底函数,并借助构造出来的基底函数研究了复模态Galerkin方法在轴向变速运动粘弹性弦线非线性振动分析中的应用．数值结果表明,复模态Galerkin方法相比实模态Galerkin方法对变系数陀螺系统有较高的收敛速度．     

热传导方程的间断Galerkin数值解法

本文对具有周期边界的热传导方程采用间断Galerkin(DG)方法给出数值求解方法,并利用傅里叶分析,对数值解进行L∞-误差估计,以一次分段多项式为例,得到半离散格式的误差估计.