On ϵ-Collocation for Pseudodifferential Equations on a Closed Curve

This paper is devoted to the approximate solution of one-dimensional pseudodifferential equations on a closed curve via spline collocation methods with variable collocation points and represents a continuation of [11]. We give necessary and sufficient conditions ensuring the L²-convergence for operators with smooth and piecewise continuous coefficients.     

On the Convergence of Adaptive Finite Element Methods

State of the art simulations in computational mechanics aim reliability and efficiency via adaptive finite element methods (AFEMs) with a posteriori error control. The a priori convergence of finite element methods is justified by the density property of the sequence of finite element spaces which essentially assumes a quasi‐uniform mesh‐refining. The advantage is guaranteed convergence for a large class of data and solutions; the disadvantage is a global mesh refinement everywhere accompanied by large computational costs. AFEMs automatically refine exclusively wherever the refinement indication suggests to do so and so violate the density property on purpose. Then, the a priori convergence of AFEMs is not guaranteed automatically and, in fact, crucially depends on algorithmic details. The advantage of AFEMs is a more effective mesh accompanied by smaller computational costs in many practical examples; the disadvantage is that the desirable error reduction property is not always guaranteed a priori. Efficient error estimators can justify a numerical approximation a posteriori and so achieve reliability. But it is not clear from the start that the adaptive mesh‐refinement will generate an accurate solution at all. This paper discusses particular versions of an AFEMs and their analyses for error reduction, energy reduction, and convergence results for linear and nonlinear problems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Full-Order Convergence of a Mixed Finite Element Method for Fourth-Order Elliptic Equations

By using a special interpolation operator and an elaborate element analysis, in this paper, we improve the classical error estimates to full order for a mixed finite element method for the fourth-order elliptic equations on the rectangular mesh. Therefore we obtain the truly optimal error estimates in view of the interpolation space for the first time.     

The Convergence of Multigrid Methods for Solving Finite Element Equations in the Presence of Singularities

1.IntroductionMultigridMethodsprovideoptimalordersolversforlinearsysternsoffiniteele-mentequationsarisingfromellipticboundaryvalueproblems.Theconvergenceofmultigridmethodswasprovedbymanya.tho.s[2-6,9-12l.AlltheseproofS,requirestrongregularitiesandquasi-uniformityofgridsl',,']-Forexample,assumingH1+oregularityandquasi-uniformtriangulations,Bank&Dupollt[3]showedaconvergencerateofo(mY),foragrowingnumbermofsmoothingstepsperlevel.Intheoptimalcasecr=1,theproblemhastobeH'-regUlar.Whentheregionhas…     

On Error Estimate of Space-Time Finite Element Methods for Parabolic Equations in a Time-Dependent Domain

We consider linear parabolic equations in a space-time domain with curved boundaries and nonhomogeneous Dirichlet boundary conditions and discuss their approximations with isoparametric space-time finite elements. A general error estimate is proved and applied to some elements of practical interest.     

双曲型积分-微分方程的有限体积元方法

本文研究了双曲型积分—微分方程的有限体积元方法，利用基于有限体积元的Ritz—Volterra投影的逼近性质，得到了半离散有限体积元解的最优阶L2，H^1，L∞和W^1,∞模误差估计．     

Finite Volume Element Methods for Two-Dimensional Three-Temperature Radiation Diffusion Equations

Two-dimensional three-temperature (2-D 3-T) radiation diffusion equations are widely used to approximately describe the evolution of radiation energy within a multi-material system and explain the exchange of energy among electrons, ions and photons. Their highly nonlinear, strong discontinuous and tightly coupled phenomena always make the numerical solution of such equations extremely challenging. In this paper, we construct two finite volume element schemes both satisfying the discrete conservation property. One of them can well preserve the positivity of analytical solutions, while the other one does not satisfy this property. To fix this defect, two as repair techniques are designed. In addition, as the numerical simulation of 2-D 3-T equations is very time consuming, we also devise a mesh adaptation algorithm to reduce the cost. Numerical results show that these new methods are practical and efficient in solving this kind of problems.     

定常Navier—Stokes方程的Petrov—Galerkin方法

研究了定常Navier－Stokes方程的四种Petrov－Galerkin有限元方法：PG1，PG2，SD和GLS．它们都是稳定的，避免了经典混合方法中必要的Babuska－Brezzi条件．给出了各种方法有限元解的存在性、唯一性和唯一解的误差估计．     

Alternating Direction Finite Volume Element Methods for Three-Dimensional Parabolic Equations

<正>This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes,one is analogous to Douglas finite difference scheme with second-order splitting error,the other two schemes have third-order splitting error,and the last one is an extended LOD scheme.The L~2 norm and H~1 semi-norm error estimates are obtained for the first scheme and second one,respectively.Finally,two numerical examples are provided to illustrate the efficiency and accuracy of the methods.     

On Stability and Convergence for Discrete-Discontinuous Finite Element Method

In this paper, we deal with the discrete-discontinuous finite element method for solving the time-dependent neutron transport equation in two-dimensional planar geometry. Its stability and convergence are proved. The numerical results are given. Compared with SN method it is of higher accuracy and superconvergence.     

Two-grid Methods for Finite Volume Element Approximations of Nonlinear Sobolev Equations

In this article, two-grid methods are studied for solving nonlinear Sobolev equation using the finite volume element method. The methods are based on one coarse grid space and one fine grid space. The nonsymmetric and nonlinear iterations are only executed on the coarse grid (with grid size H), and the fine grid solution (with grid size h) can be obtained in a single symmetric and linear step. The optimal H¹ error estimates are presented for the proposed methods, which show that the two-grid methods achieve optimal approximation as long as the mesh sizes satisfy h = 𝒪(H³|ln H|). As a result, solving such a large class of nonlinear Sobolev equations will not be much more difficult than solving one linearized equation.     

Constraint-Preserving Hybrid Finite Element Methods for Maxwell’s Equations

Foundations of Computational Mathematics - Maxwell’s equations describe the evolution of electromagnetic fields, together with constraints on the divergence of the magnetic and electric flux...     

On Finite Element Methods for Inhomogeneous Dielectric Waveguides

We investigate the problem of computing electromagnetic guided waves in a closed,inhomogeneous, pillared three-dimensional waveguide at a given frequency. The problem is formulated as a generalized eigenvalue problem. By modifying the sesquilinear form associated with the eigenvalue problem, we provide a new convergence analysis for the finite element approximations. Numerical results are reported to illustrate the performance of the method.     

Sobolev-Volterra投影与积分微分方程有限元数值分析

本文提出一类称之为Sobolev-Volterra投影的有限元投影,研究了有关性质并将之应用于伪抛物型积分微分方程有限元方法、伪双曲型积分微分方程有限元方法以及三维伪双曲型积分微分方程交替方向有限元方法的数值分析.     

Convergence Analysis for the Iterated Defect Correction Scheme of Finite Element Methods on Rectangle Grids

This paper develops a new method to analyze convergence of the iterated defect correction scheme of finite element methods on rectangular grids in both two and three dimensions. The main idea is to formulate energy inner products and energy (semi)norms into matrix forms. Then, two constants of two key inequalities involved are min and max eigenvalues of two associated generalized eigenvalue problems, respectively. Local versions on the element level of these two generalized eigenvalue problems are exactly solved to obtain sharp (lower) upper bounds of these two constants. This and some essential observations for iterated solutions establish convergence in 2D and the monotone decreasing property in 3D. For two dimensions the results herein improve those in literature; for three dimensions the results herein are new. Numerical results are presented to examine theoretical results.     

Superconvergence of H 1-Galerkin Mixed Finite Element Methods for Second-Order Elliptic Equations

We derive superconvergence result for H ¹-Galerkin mixed finite element method for second-order elliptic equations over rectangular partitions. Compared to standard mixed finite element procedure, the method is not subject to the Ladyzhenskaya–Bab?ska–Brezzi (LBB) condition and the approximating finite element spaces are allowed to be of different polynomial degrees. Superconvergence estimate of order 𝒪(h ^k+3), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas elements, is established for the flux via a postprocessing technique.     

On the Exponential Convergence of Spline Approximation Methods for Wiener-Hopf Equations

This paper deals with the study of some new multidimensional operators of fractional calculus. These operators are considered in certain spaces of test and generalized functions. Some mapping properties and product rules are obtained.     

Convergence of Legendre Methods for Navier-Stokes Equations

1.IntroductionInthispaPerwestudyspectraltyPeofmethodsbasedonLegendrepo1ynomials.Weprovestabilityandconvergenceresultsforthesemethodsusingenergyestimates-TheconvergenceresultsweobtainedaxenearlyoptimalinthesensethattheerrorestimatesforthenumericalsolutionisofthesameorderastheerrorestimatesinaPproalmationtheoryl7,14].Atrivialconsequenceisthatthesemethodsareindeedspectrallyaccurate.Spectralmethodshavebeenusedqulteextensivelyinthepasttwodecades-Be-causeoftheirhighresolutionpower,thesemethodsrec…