Adaptive computation of smallest eigenvalues of self‐adjoint elliptic partial differential equations

We consider a new adaptive finite element (AFEM) algorithm for self‐adjoint elliptic PDE eigenvalue problems. In contrast to other approaches we incorporate the inexact solutions of the resulting finite‐dimensional algebraic eigenvalue problems into the adaptation process. In this way we can balance the costs of the adaptive refinement of the mesh with the costs for the iterative eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the eigenvalue solver and roundoff errors, and use these for the adaptation process. We show that it is also possible to restrict to very few iterations of a Krylov subspace solver for the eigenvalue problem on coarse meshes. Several examples are presented to show that this new approach achieves much better complexity than the previous AFEM approaches which assume that the algebraic eigenvalue problem is solved to full accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

On the numerical approach of the enthalpy method for the Stefan problem

In this article an error bound is derived for a piecewise linear finite element approximation of an enthalpy formulation of the Stefan problem; we have analyzed a semidiscrete Galerkin approximation and completely discrete scheme based on the backward Euler method and a linearized scheme is given and its convergence is also proved. A second‐order error estimates are derived for the Crank‐Nicolson Galerkin method. In the second part, a new class of finite difference schemes is proposed. Our approach is to introduce a new variable and transform the given equation into an equivalent system of equations. Then, we prove that the difference scheme is second order convergent. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

A numerical technique for solving fractional variational problems

This paper presents an accurate numerical method for solving a class of fractional variational problems (FVPs). The fractional derivative in these problems is in the Caputo sense. The proposed method is called fractional Chebyshev finite difference method. In this technique, we approximate FVPs and end up with a finite‐dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The Caputo fractional derivative is replaced by a difference quotient and the integral by a finite sum. The fractional derivative approximation using Clenshaw and Curtis formula introduced here, along with Clenshaw and Curtis procedure for the numerical integration of a non‐singular functions and the Rayleigh–Ritz method for the constrained extremum, is considered. By this method, the given problem is reduced to the problem for solving a system of algebraic equations, and by solving this system, we obtain the solution of FVPs. Special attention is given to study the convergence analysis and evaluate an error upper bound of the obtained approximate formula. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique. A comparison with another method is given. Copyright © 2012 John Wiley & Sons, Ltd.     

A characteristic-mixed finite element method for time-dependent convection-diffusion optimal control problem

In this paper, we examine the method of characteristic-mixed finite element for the approximation of convex optimal control problem governed by time-dependent convection-diffusion equations with control constraints. For the discretization of the state equation, the characteristic finite element is used for the approximation of the material derivative term (i.e., the time derivative term plus the convection term), and the lowest-order Raviart-Thomas mixed element is applied for the approximation of the diffusion term. We derive some a priori error estimates for both the state and control approximations.     

Explicit error estimate for the nonconforming wilson's element

In this article, we study the explicit expressions of the constants in the error estimate of the nonconforming finite element method. We explicitly obtain the approximation error estimate and the consistency error estimate for the Wilson's element without the regular assumption, respectively, which implies the final finite element error estimate. Such explicit a priori error estimates can be used as computable error bounds.     

解三维抛物型方程的一个高精度显式差分格式

提出了求解三维抛物型方程的一个高精度显式差分格式.首先,推导了一个特殊节点处一阶偏导数(■u)/(■/t)的一个差分近似表达式,利用待定系数法构造了一个显式差分格式,通过选取适当的参数使格式的截断误差在空间层上达到了四阶精度和在时间层上达到了三阶精度.然后,利用Fourier分析法证明了当r1/6时,差分格式是稳定的.最后,通过数值试验比较了差分格式的解与精确解的区别,结果说明了差分格式的有效性.     

一类半线性反应对流扩散模型的特征差分方法和分析

1.引 言如下形式的半线性反应对流扩散方程组分别在生命科学、化学和环境科学中,有大量的应用模型[1-3].其中文献[2-6]分别讨论了方程组(1.1)的各种特殊模型的定性性质.文献[6]讨论了一类线性模型的流线扩散有限元分析.作者在文[7]中,分别利用标准有限元方法和交替方向有限元方法,对(1.1)的一些特殊情形作了数值分析.     

Closed-form expressions for the finite difference approximations of first and higher derivatives based on Taylor series

Numerical differentiation formulas based on interpolating polynomials, operators and lozenge diagrams can be simplified to one of the finite difference approximations based on Taylor series. In this paper, we have presented closed-form expressions of these approximations of arbitrary order for first and higher derivatives. A comparison of the three types of approximations is given with an ideal digital differentiator by comparing their frequency responses. The comparison reveals that the central difference approximations can be used as digital differentiators, because they do not introduce any phase distortion and their amplitude response is closer to that of an ideal differentiator. It is also observed that central difference approximations are in fact the same as maximally flat digital differentiators. In the appendix, a computer program, written in MATHEMATICA is presented, which can give the approximation of any order to the derivative of a function at a certain mesh point.     

本征值有限元近似的一个抽象误差估计式

设T：LZ（fi）MLZ（fi）是自共轭全连续算子，SgCLZ（fi）是分片。次有限元空间，几：LZ、St是有限秩自共轭算子，IITh-Tllo、0（h、0）．考虑本征值问题：及其近似用（·，·）和DD·D【。·分别表示h（m中内积和范数·ID·卜F表示w认｝（m中范数，简记D卜队。为D卜卜·因为T自共轭全连续，所以它有可数无穷个本征值h，人，...人，...．其相应的本征函数（2丹构成完全标准直交系，所以VZELZ（m设几的本征值为A。l，汕。，...，汕n，相应的本征函数为如山，卜则。二1·不失一般性，可EitL。设tik一大干二＞，E是到Ah对应的本…     

NUMERICAL SOLUTION OF THE SPACE FRACTIONAL DIFFERENTIAL EQUATION

In this paper, a space fractional differential equation is considered. The equation is obtained from the parabolic equation containing advection, diffusion and reaction terms by replacing the second order derivative in space by a fractional derivative in space of order. An implicit finite difference approximation for this equation is presented. The stability and convergence of the finite difference approximation are proved. A fractional-order method of lines is also presented. Finally, some numerical results are given.     

On the numerical approximation of unstable minimal surfaces with polygonal boundaries

Summary. This work is concerned with the approximation and the numerical computation of polygonal minimal surfaces in . Polygonal minimal surfaces correspond to the critical points of Shiffman's function . Since this function is analytic, polygonal minimal surfaces can be characterized by means of the second derivative of . We present a finite element approximation of quasiminimal surfaces together with an error estimate. In this way we obtain discrete approximations of and of . In particular we prove that the discrete functions converge uniformly on certain compact subsets. This will be the main tool for proving existence and convergence of discrete minimal surfaces in neighbourhoods of non-degenerate minimal surfaces. In the numerical part of this paper we compute numerical approximations of polygonal minimal surfaces by use of Newton's method applied to . Received October 27, 1994     

Discontinuous Galerkin finite element method for a nonlinear boundary value problem

In this paper, we investigate the a priori and a posteriori error estimates for the discontinuous Galerkin finite element approximation to a regularization version of the variational inequality of the second kind. We show the optimal error estimates in the DG-norm （stronger than the H1 norm） and the L2 norm, respectively. Furthermore, some residual-based a posteriori error estimators are established which provide global upper bounds and local lower bounds on the discretization error. These a posteriori analysis results can be applied to develop the adaptive DG methods.     

Numerical treatment for solving fractional Riccati differential equation

This paper presents an accurate numerical method for solving fractional Riccati differential equation (FRDE). The proposed method so called fractional Chebyshev finite difference method (FCheb-FDM). In this technique, we approximate FRDE with a finite dimensional problem. The method is based on the combination of the useful properties of Chebyshev polynomials approximation and finite difference method. The Caputo fractional derivative is replaced by a difference quotient and the integral by a finite sum. By this method the given problem is reduced to a problem for solving a system of algebraic equations, and by solving this system, we obtain the solution of FRDE. Special attention is given to study the convergence analysis and estimate an error upper bound of the obtained approximate formula. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique.     

对流扩散方程一类改进的特征线修正有限元方法

１引言在地下水污染,地下渗流驱动,核污染,半导体等问题的数值模拟中,均涉及抛物型对流扩散方程（或方程组）的数值求解问题．这些对流扩散型偏微分方程（或方程组）具有共同的特点：对流的影响远大于扩散的影响,即对流占优性,对流占优性给问题的数值求解带来许多困难,因此对流占优问题的有效数值解法一直是计算数学中重要的研究内容．用通常的差分法或有限元法进行数值求解将出现数值振荡．为了克服数值振荡,提出各种迎风方法和修正的特征方法并在这些问题上得到成功的实际应用、８０年代,Ｄｏｕｇｌａｓ和Ｒｕｓｓｅｌｌ［２］等…     

A posteriori error analysis for nonconforming approximations of an anisotropic elliptic problem

We develop in this article an a posteriori error estimator for the P₁‐nonconforming finite element approximation, for a diffusion‐reaction equation. We adopt the error in a constitutive law approach in two and three dimensional space, for not necessary piecewise constant data of problems. The efficiency and the reliability of our estimators are proved, neither Helmholtz decomposition of the error nor saturation assumption. The constants are explicitly given, which prove the robustness of these estimators. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 950–976, 2015     

Expansion Formulas in Terms of Integer-Order Derivatives for the Hadamard Fractional Integral and Derivative

We obtain series expansion formulas for the Hadamard fractional integral and fractional derivative of a smooth function. When considering finite sums only, an upper bound for the error is given. Numerical simulations show the efficiency of the approximation method.     

Optimal finite difference grids and rational approximations of the square root I. Elliptic problems

The main objective of this paper is optimization of second‐order finite difference schemes for elliptic equations, in particular, for equations with singular solutions and exterior problems. A model problem corresponding to the Laplace equation on a semi‐infinite strip is considered. The boundary impedance (Neumann‐to‐Dirichlet map) is computed as the square root of an operator using the standard three‐point finite difference scheme with optimally chosen variable steps. The finite difference approximation of the boundary impedance for data of given smoothness is the problem of rational approximation of the square root on the operator's spectrum. We have implemented Zolotarev's optimal rational approx‐imant obtained in terms of elliptic functions. We have also found that a geometrical progression of the grid steps with optimally chosen parameters is almost as good as the optimal approximant. For bounded operators it increases from second to exponential the convergence order of the finite difference impedance with the convergence rate proportional to the inverse of the logarithm of the condition number. For the case of unbounded operators in Sobolev spaces associated with elliptic equations, the error decays as the exponential of the square root of the mesh dimension. As an example, we numerically compute the Green function on the boundary for the Laplace equation. Some features of the optimal grid obtained for the Laplace equation remain valid for more general elliptic problems with variable coefficients. © 2000 John Wiley & Sons, Inc.     

An a priori error analysis for the coupling of local discontinuous Galerkin and boundary element methods

In this paper we analyze the coupling of local discontinuous Galerkin (LDG) and boundary element methods as applied to linear exterior boundary value problems in the plane. As a model problem we consider a Poisson equation in an annular polygonal domain coupled with a Laplace equation in the surrounding unbounded exterior region. The technique resembles the usual coupling of finite elements and boundary elements, but the corresponding analysis becomes quite different. In particular, in order to deal with the weak continuity of the traces at the interface boundary, we need to define a mortar-type auxiliary unknown representing an interior approximation of the normal derivative. We prove the stability of the resulting discrete scheme with respect to a mesh-dependent norm and derive a Strang-type estimate for the associated error. Finally, we apply local and global approximation properties of the subspaces involved to obtain the a priori error estimate in the energy norm.

     

Adaptive solution of elliptic PDE-eigenvalue problems.

In this paper we introduce a new adaptive algorithm (AFEMLA) for elliptic PDE-eigenvalue problems. In contrast to other approaches the algebraic eigenvalue problem does not have to be solved to full accuracy. We incorporate the iterative solution of the resulting finite dimensional algebraic eigenvalue problems in the adaptation process in order to balance the cost with the costs for the iterative eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the eigenvalue solver and roundoff errors. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Roundoff error analysis of algorithms based on Krylov subspace methods

We study the roundoff error propagation in an algorithm which computes the orthonormal basis of a Krylov subspace with Householder orthonormal matrices. Moreover, we analyze special implementations of the classical GMRES algorithm, and of the Full Orthogonalization Method. These techniques approximate the solution of a large sparse linear system of equations on a sequence of Krylov subspaces of small dimension. The roundoff error analyses show upper bounds for the error affecting the computed approximated solutions.This work was carried out with the financial contribution of the Human Capital and Mobility Programme of the European Union grant ERB4050PL921378.