Spectral computations: from operators to matrices

In this work we will address the problem of finding spectral values and bases for the corresponding spectral subspaces for a bounded operator on a Banach space. We will make a bridge between the spectral problem for the continuous operator and the computation of the eigenpairs of a matrix. This approximate problem results first from an approximation by a sequence of continuous operators with finite rank, followed by the reduction to a spectral problem for an operator whose domain as well as range are finite dimensional. Some discussion on defect correction procedures will be also presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonconforming Mixed Finite Element Method for Time-Dependent Maxwell's Equations with ABC

In this paper, a nonconforming mixed finite element method (FEM) is presented to approximate time-dependent Maxwell's equations in a three-dimensional bounded domain with absorbing boundary conditions (ABC). By employing traditional variational formula, instead of adding penalty terms, we show that the discrete scheme is robust. Meanwhile, with the help of the element's typical properties and derivative transfer skills, the convergence analysis and error estimates for semi-discrete and backward Euler fully-discrete schemes are given, respectively. Numerical tests show the validity of the proposed method.     

Error estimates for the finite element approximation of linear elastic equations in an unbounded domain

In this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain. The finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error estimates show how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. A numerical example for Navier equations outside a circle in the plane is presented. Numerical results demonstrate the performance of our error estimates.

     

Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations

In this paper, some reduced finite difference schemes based on a proper orthogonal decomposition (POD) technique for parabolic equations are derived. Also the error estimates between the POD approximate solutions of the reduced finite difference schemes and the exact solutions for parabolic equations are established. It is shown by considering the results of two numerical examples that the numerical results are consistent with theoretical conclusions. Moreover, it is also shown that the POD reduced finite difference schemes are feasible and efficient.     

Bases of spaces of matrices satisfying rank conditions

Motivated by the problem concerning the existence of non-singular bilinear maps, vector spaces of matrices consisting of matrices with rank bounded below are investigated. It is shown that bases for such spaces of maximum dimension can be chosen in such a way to consist of matrices of the minimal rank. An estimate of the ranks of matrices in particular types of bases for maximal such spaces is also given. This extends previously known results which were valid only in the case of spaces consisting of matrices of rank not equal to one.     

Boundary layer approximate approximations and cubature of potentials in domains

In this article we present a new approach to the computation of volume potentials over bounded domains, which is based on the quasi‐interpolation of the density by almost locally supported basis functions for which the corresponding volume potentials are known. The quasi‐interpolant is a linear combination of the basis function with shifted and scaled arguments and with coefficients explicitly given by the point values of the density. Thus, the approach results in semi‐analytic cubature formulae for volume potentials, which prove to be high order approximations of the integrals. It is based on multi‐resolution schemes for accurate approximations up to the boundary by applying approximate refinement equations of the basis functions and iterative approximations on finer grids. We obtain asymptotic error estimates for the quasi‐interpolation and corresponding cubature formulae and provide some numerical examples. This revised version was published online in June 2006 with corrections to the Cover Date.     

SPECTRAL AND PSEUDOSPECTRAL APPROXIMATIONS IN TIME FOR PARABOLIC EQUATIONS

1.IntroductionInrecentyears)ithasbeenshownthatspectralmethodsareveryusefultosolvepanicaldifferentialequations.Spectralmethods,inwhichtheapproximatesolutionisapolyllomialofhiglldegree,areknowntobeveryaccuratewhenthesolutiontobeal)proximatedisverysmooth(see…     

Couette-Taylor流的谱Galerkin逼近

利用谱方法对轴对称的旋转圆柱间的Couette-Taulor流进行数值模拟．首先给出Navier-Stokes方程流函数形式,利用Couette流把边界条件齐次化．其次给出Stokes算子的特征函数的解析表达式,证明其正交性,并对特征值进行估计．最后利用Stokes算子的特征函数作为逼近子空间的基函数,给出谱Galerkin逼近方程的表达式．证明了Navier-Stokes方程非奇异解的谱Galerkin逼近的存在性、唯一性和收敛性,给出了解谱Galerkin逼近的误差估计,并展示了数值计算结果．     

FINITE ELEMENT METHODS FOR SOBOLEV EQUATIONS

A new high-order time-stepping finite element method based upon the high-order numerical integration formula is formulated for Sobolev equations, whose computations consist of an iteration procedure coupled with a system of two elliptic equations. The optimal and superconvergence error estimates for this new method are derived both in space and in time. Also, a class of new error estimates of convergence and superconvergence for the time-continuous finite element method is demonstrated in which there are no time derivatives of the exact solution involved, such that these estimates can be bounded by the norms of the known data. Moreover, some useful a-posteriori error estimators are given on the basis of the superconvergence estimates.     

Convergence Analysis of a Coupled Finite Element and Spectral Method in Acoustic Scattering

In this paper we provide convergence estimates for a coupledfinite element and spectral method for approximating time harmonicacoustic scattering by a bounded inhomogeneity. We use a finiteelement procedure to approximate the near field in the regionof the inhomogeneity and couple this finite element solutionto a Hankel function expansion away from the inhomogeneity.We provide error estimates that take into account both the finiteelement and the series discretization errors. We suggest a methodfor implementing the algorithm using the preconditioned conjugategradient method, and provide some numerical results.     

A Low Order Nonconforming Mixed Finite Element Method for Non-Stationary Incompressible Magnetohydrodynamics System

A low order nonconforming mixed finite element method (FEM) is established for the fully coupled non-stationary incompressible magnetohydrodynamics (MHD) problem in a bounded domain in 3D. The lowest order finite elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field and the magnetic field, in which the hydrodynamic unknowns are approximated by inf-sup stable finite element pairs and the magnetic field by $H^1(\Omega)$-conforming finite elements, respectively. The existence and uniqueness of the approximate solutions are shown. Optimal order error estimates of $L^2(H^1)$-norm for the velocity field, $L^2(L^2)$-norm for the pressure and the broken $L^2(H^1)$-norm for the magnetic field are derived.     

Approximate solutions of problems involving normal operators

The spectral theory for unbounded normal operators is used to develop a systematic method of approximating functions of operators with other, more easily computable functions, leading to a priori error estimates in the operator norm. In particular, polynomial approximations are obtained for resolvents and semigroups in terms of inverses and resolvents, respectively.     

Convergence of Fourier spectral method for resonant long-short nonlinear wave interaction

In this paper, the evolution equations with nonlinear term describing the resonance interaction between the long wave and the short wave are studied. The semi-discrete and fully discrete Crank-Nicholson Fourier spectral schemes are given. An energy estimation method is used to obtain error estimates for the approximate solutions. The numerical results obtained are compared with exact solution and found to be in good agreement.     

On the finite element approximation of incompressible flows of an electrically conducting fluid

We consider the finite element approximation of incompressible flows field of an electrically conducing fluid in the presence of a magnetic where it is assumed that this field is prescribed. A weak form is chosen that is similar in some respects to a weak form used by many authors for the Navier-Stokes equations. Existence and uniqueness results are presented for the weak problem. A finite element Algorithm is given for the approximate solution of the weak problem and error estimates are derived.     

发展方程初边值问题的高阶差分-边界积分方程法及误差分析

本文结合差分方法与边界积分方程方法,提出并研究了一类新的求解发展型方程初边值问题的高阶差分与边界积分方程耦合数值方法.对于有界区域问题与无界区域问题给出了数值计算格式及其误差的先验估计.     

Error estimates of the spectral collocation method for the Ginzburg–Landau equation coupled with the Benjamin–Bona–Mahony equation

In this paper, we consider the spectral collocation method for the Ginzburg–Landau equation coupled with the Benjamin–Bona–Mahony equation. Semidiscrete and fully discrete spectral collocation schemes are given. In the fully discrete case, a three-level spectral collocation scheme is considered. An energy estimation method is used to obtain error estimates for the approximate solutions. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.     

An efficient spectral‐Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrödinger equations

In this paper, we propose an efficient spectral‐Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrödinger equations. Firstly, we carry out a truncation from a three‐dimensional unbounded domain to a bounded spherical domain. By using spherical coordinate transformation and spherical harmonic expansion, we transform the original problem into a series of one‐dimensional eigenvalue problem that can be solved effectively. Secondly, we introduce a weighted Sobolev space to treat the singularity in the effective potential. Using the property of orthogonal polynomials in weighted Sobolev space, the error estimate for the approximate eigenvalues and corresponding eigenfunctions are proved. Error estimates show that our numerical method can achieve spectral accuracy for approximate eigenvalues and eigenfunctions. Finally, we give some numerical examples to demonstrate the efficiency of our algorithms and the correctness of the theoretical results.     

Multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations

This study develops a novel multiscale computational method for thermoelastic problems of composite materials with orthogonal periodic configurations. Firstly, the multiscale asymptotic analysis for these multiscale problems is given successfully, and the formal second-order two-scale approximate solutions for these multiscale problems are constructed based on the above-mentioned analysis. Then, the error estimates for the second-order two-scale (SOTS) solutions are obtained. Furthermore, the corresponding SOTS numerical algorithm based on finite element method (FEM) is brought forward in details. Finally, some numerical examples are presented to verify the feasibility and effectiveness of our multiscale computational method. Moreover, our multiscale computational method can accurately capture the local thermoelastic responses in composite block structure, plate, cylindrical and doubly-curved shallow shells.     

Approximation of Infinite Boundary Condition and Its Application to Finite Element Methods

The exterior boundary value problems of Laplace equation and linear elastic equations are considered. A series of approximate infinite boundary conditions are given. Then the original problem is reduced to a boundary value problem on a bounded domain. The finite element approximation of this problem and its error estimate are obtained. Finally, a numerical example shows that this method is very effective.     

The Modified Upwind Finite Difference Fractional Steps Method for Compressible Two-phase Displacement Problem

For compressible two-phase displacement problem,the modified upwind finite difference fractionalsteps schemes are put forward.Some techniques,such as calculus of variations,commutative law of multiplicationof difference operators,decomposition of high order difference operators,the theory of prior estimates and tech-niques are used.Optimal order estimates in L~2 norm are derived for the error in the approximate solution.Thismethod has already been applied to the numerical simulation of seawater intrusion and migration-accumulationof oil resources.