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1.
Summary. We derive pointwise weighted error estimates for a semidiscrete finite element method applied to parabolic equations. The results extend those obtained by A.H. Schatz for stationary elliptic problems. In particular, they show that the error is more localized for higher order elements.
Mathematics Subject Classification (2000): 65N30 相似文献
2.
《Numerical Functional Analysis & Optimization》2013,34(7-8):891-909
ABSTRACT A posteriori error estimates for semidiscrete finite element methods for a nonlinear parabolic initial-boundary value problem are considered. The error estimates are obtained by solving local parabolic or elliptic equations for corrections to the solution on each element. The convergence results improve previous results where unnecessary assumptions are imposed on the approximate solution and the elliptic projection of the exact solution. 相似文献
3.
In this paper, we discuss with guaranteed a priori and a posteriori error estimates of finite element approximations for not necessarily coercive linear second order Dirichlet problems. Here, ‘guaranteed’ means we can get the error bounds in which all constants included are explicitly given or represented as a numerically computable form. Using the invertibility condition of concerning elliptic operator, guaranteed a priori and a posteriori error estimates are formulated. This kind of estimates plays essential and important roles in the numerical verification of solutions for nonlinear elliptic problems. Several numerical examples that confirm the actual effectiveness of the method are presented. 相似文献
4.
Elliptic reconstruction and a posteriori error estimates for parabolic optimal control problems
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In this article, a semidiscrete finite element method for parabolic optimal control problems is investigate. By using elliptic reconstruction, a posteriori error estimates for finite element discretizations of optimal control problem governed by parabolic equations with integral constraints are derived. 相似文献
5.
Mixed methods for fourth-order elliptic and parabolic problems using radial basis functions 总被引:1,自引:0,他引:1
Jichun Li 《Advances in Computational Mathematics》2005,23(1-2):21-30
By extending Wendlands meshless Galerkin methods using RBFs, we develop mixed methods for solving fourth-order elliptic and parabolic problems by using RBFs. Similar error estimates as classical mixed finite element methods are proved.
AMS subject classification 35G15, 65N12 相似文献
6.
Hans-Jürgen Reinhardt 《Applied Numerical Mathematics》1985,1(2):145-176
Two types of singularly perturbed, linear partial differential equations are considered, namely time dependent convection-diffusion problems and a two-dimensional elliptic equation having a first order unperturbed operator. Finite element approximations are constructed via modifications of classical methods of lines. The main purpose of the present work is to establish a-posteriori estimates for the error between the solutions of the finite element methods and the boundary value problems arising from the line methods. The different types of equations, parabolic and elliptic ones, and distinct directions of the lines in the elliptic case require different techniques in order to derive estimates of the desired form. These realistic, local a-posteriori error estimates may be used as a basis for adaptive computations refining the mesh automatically on every discrete line. 相似文献
7.
刘蕴贤 《数学物理学报(A辑)》2005,25(4):433-444
该文用交替方向有限元方法求解半导体问题的Energy Trans port (ET)模型。对模型中椭圆型的电子位势方程采用交替方向迭代法,对流占优扩散的电子浓度和空穴浓度方程采用特征交替方向有限元方法,热传导方程利用Patch逼近采用交替方向有限元方法求解。利用微分方程的先验估计理论和技巧,分别得到了椭圆型方程和抛物型方程的最优H+1和L+2误差估计。 相似文献
8.
We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L 2(Ω) and H 1(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose. 相似文献
9.
In this paper we investigate a space-time finite element approximation of parabolic optimal control problems. The first order optimality conditions are transformed into an elliptic equation of fourth order in space and second order in time involving only the state or the adjoint state in the space-time domain. We derive a priori and a posteriori error estimates for the time discretization of the state and the adjoint state. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
10.
Lothar Collatz 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1958,9(2):118-128
Summary Types of boundary value problems of partial differential equations for infinite domains are discussed which can easily be transformed in such a manner as to allow estimations of error (for approximate solutions) similar to the boundary maximum principle. First, second und third boundary value problems for the outer domain of linear elliptic and certain linear and nonlinear parabolic differential equations are examined. For elliptic differential equations one of the results is that the secound boundary value problem for more than two dimensions can be included. The estimates of the paper can thus be applied to problems of flow around some object, not in the case of two but of three dimensions. This is in a certain sense a counterpart to the conformal mappings method which is successful for two but not for three dimensions. Numerical examples show that estimations of error can easily be carried out. 相似文献
11.
Bhupen Deka 《Journal of Computational and Applied Mathematics》2010,234(2):605-612
The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L2 and H1 norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings. 相似文献
12.
Peter K. Moore 《Numerische Mathematik》2003,94(2):367-401
Summary. Hp-adaptive finite element codes require methods for estimating the error at several spatial orders and for interpolating solutions
between grids. Lobatto polynomial-based techniques are presented for both. An interpolation error-based error estimation strategy
for a posteriori error estimates is generalized to yield asymptotically exact error estimates one order higher than the computed solution.
The estimates involve high-order derivatives of the solution that must be approximated from the computed solution. Differentiating
a ``Taylor-like' series for error in the Lobatto interpolant and using the weak form of the equations yields the correct
derivative approximations. This leads to a more robust order selection strategy. Interpolation between grids is done over
each element using the Lobatto interpolating polynomial. Explicit formulas for the inverse of the resulting Lobatto interpolation
matrices are given. Computational results illustrate the theory.
Received June 25, 2001 / Revised version received February 12, 2002 / Published online October 29, 2002
Mathematics Subject Classification (1991): 65M15,65M20,65M60
This research was partially supported by NSF Grant #DMS-0196108. 相似文献
13.
Summary Consider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewiseP
th
-degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degreep+1 that vanish at element ends. Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 90-0194; by the U.S. Army Research Office under Contract Number DAAL03-91-G-0215; and by the National Science Foundation under Institutional Infrastructure Grant Number CDA-8805910 相似文献
14.
Summary. Moving from the anisotropic interpolation error estimates derived in [12], we provide here both a-priori and a-posteriori
estimates for a generic elliptic problem. The a-priori result is deduced by following the standard finite element theory.
For the a-posteriori estimate, the analysis extends to anisotropic meshes the theory presented in [3–5]. Numerical test-cases
validate the derived results.
Received July 22, 2001 / Revised version received March 20, 2002 / Published online July 18, 2002
Mathematics Subject Classification (1991): 65N15, 65N50 相似文献
15.
We study the semidiscrete Galerkin approximation of a stochastic parabolic partial differential equation forced by an additive space-time noise. The discretization in space is done by a piecewise linear finite element method. The space-time noise is approximated by using the generalized L2 projection operator. Optimal strong convergence error estimates in the L2 and
norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The error estimates are applicable in the multi-dimensional case.
AMS subject classification (2000) 65M, 60H15, 65C30, 65M65.Received April 2004. Revised September 2004. Communicated by Anders Szepessy. 相似文献
16.
In this paper we study the residual type a posteriori error estimates for general elliptic (not necessarily symmetric) eigenvalue problems. We present estimates for approximations of semisimple eigenvalues and associated eigenvectors. In particular, we obtain the following new results: 1) An error representation formula which we use to reduce the analysis of the eigenvalue problem to the analysis of the associated source problem; 2) A local lower bound for the error of an approximate finite element eigenfunction in a neighborhood of a given mesh element T. 相似文献
17.
陈艳萍 《数学物理学报(B辑英文版)》1999,19(4):449
1IntroductionThebase0fadaPtivecomputing0ffiniteelementmethodisap0steri0rierr0restimates.I.Babuskaisthepioneerinthisfields.Manytechniquesaredevel0pedtoobtainaposteri0rierrorestimators.See[1-3,7-8,19-201.Theyaremainlybased0nthejumps0fthederiva-tivesontheboundariesoftl1eelel11elltandtheresidualintheelemellts.Recelltresultssh0wthatthereareveryclosedrelatiollsbetweellasymptoticexactap0steri0rierrorestimatesandsuperc0nvergence-SeealsoQ.Linetal.[11-13],andChen-Huang['].Therehasbeenmuchprogressill… 相似文献
18.
We derive residual based a posteriori error estimates for parabolic problems on mixed form solved using Raviart–Thomas–Nedelec
finite elements in space and backward Euler in time. The error norm considered is the flux part of the energy, i.e. weighted
L
2(Ω) norm integrated over time. In order to get an optimal order bound, an elementwise computable post-processed approximation
of the scalar variable needs to be used. This is a common technique used for elliptic problems. The final bound consists of
terms, capturing the spatial discretization error and the time discretization error and can be used to drive an adaptive algorithm. 相似文献
19.
Summary.
In an abstract framework we present a formalism which
specifies the notions of consistency and stability of
Petrov-Galerkin
methods used to approximate nonlinear problems which are, in many
practical situations, strongly nonlinear elliptic problems. This
formalism gives rise to a priori and a posteriori error estimates which
can be used for the refinement of the mesh in adaptive finite element
methods applied to elliptic nonlinear problems. This theory is
illustrated with the example: in a two
dimensional domain with Dirichlet boundary conditions.
Received June 10, 1992 / Revised version received February
28, 1994 相似文献
20.
Chin-Yun Chen 《BIT Numerical Mathematics》2007,47(2):297-312
Peano kernels are critical for the error estimates of numerical approximation of linear functionals. In the literature, considerable
efforts have been devoted to the numerical computation or estimation of Peano constants, however, their discussions have mainly
focused on the Peano kernels of some specific higher orders related to the degrees of the underlying error functionals. Limiting
to such higher orders either requires certain smoothness of the approximated functions, which is not always fulfilled, or
is not always the optimal choice for error estimates, even if the approximated functions are sufficiently smooth. Practical
considerations in computing Peano constants for the full range of orders are given in this paper. Numerical examples are also
given to demonstrate that much better error bounds can be derived while Peano constants of lower orders are considered.
AMS subject classification (2000) 41-04, 41A44, 41A55, 41A80, 65-04, 65A05, 65D20, 65D30, 65G20, 65G30 相似文献