An adaptive time stepping method with efficient error control for second-order evolution problems

This work is concerned with time stepping fnite element methods for abstract second order evolution problems.We derive optimal order a posteriori error estimates and a posteriori nodal superconvergence error estimates using the energy approach and the duality argument.With the help of the a posteriori error estimator developed in this work,we will further propose an adaptive time stepping strategy.A number of numerical experiments are performed to illustrate the reliability and efciency of the a posteriori error estimates and to assess the efectiveness of the proposed adaptive time stepping method.     

A posteriori error estimates for mixed FEM in elasticity

A residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value problems in linear, planar elasticity. The proof of the reliability of the estimator is based on Helmholtz type decompositions of the error in the stress variable and a duality argument for the error in the displacements. The efficiency follows from inverse estimates. The constants in both estimates are independent of the Lamé constant , and so locking phenomena for are properly indicated. The analysis justifies a new adaptive algorithm for automatic mesh–refinement. Received July 17, 1997     

A priori error estimates for approximate solutions to convex conservation laws

Summary. We introduce a new technique for proving a priori error estimates between the entropy weak solution of a scalar conservation law and a finite–difference approximation calculated with the scheme of Engquist-Osher, Lax-Friedrichs, or Godunov. This technique is a discrete counterpart of the duality technique introduced by Tadmor [SIAM J. Numer. Anal. 1991]. The error is related to the consistency error of cell averages of the entropy weak solution. This consistency error can be estimated by exploiting a regularity structure of the entropy weak solution. One ends up with optimal error estimates. Received December 21, 2001 / Revised version received February 18, 2002 / Published online June 17, 2002     

A posteriori error estimation for nonlinear variational problems

We introduce a new modus operandi for a posteriori error estimation for nonlinear (and linear) variational problems based on the duality theory of the calculus of variations. We derive what we call duality error estimates and show that they yield computable a posteriori error estimates without directly solving the dual problem.     

A posteriori error estimation for nonlinear variational problems by duality theory

In this paper, we use the duality theory of the calculus of variations to derive a posteriori error estimates. We obtain a general form of this (duality) error estimate and show that the known classes of a posteriori error estimates are its particular cases. Bibliography: 21 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 201–214. Translated by S. I. Repin.     

Error estimates for obstacle problems of higher order

For obstacle problems of higher order involving power growth functionals, a posteriori error estimates using methods in duality theory are proved. These estimates can be viewed as a reliable measure for the deviation of an approximation from the exact solution, which is independent of the concrete numerical scheme under consideration. Bibliography: 11 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 5–18.     

A posteriori error estimates of mixed DG finite element methods for linear parabolic equations

In this article, we analyse a posteriori error estimates of mixed finite element discretizations for linear parabolic equations. The space discretization is done using the order λ?≥?1 Raviart–Thomas mixed finite elements, whereas the time discretization is based on discontinuous Galerkin (DG) methods (r?≥?1). Using the duality argument, we derive a posteriori l ^∞(L ²) error estimates for the scalar function, assuming that only the underlying mesh is static.     

The element‐free Galerkin method for a quasistatic contact problem with the Tresca friction in elastic materials

This paper is proposed for the error estimates of the element‐free Galerkin method for a quasistatic contact problem with the Tresca friction. The penalty method is used to impose the clamped boundary conditions. The duality algorithm is also given to deal with the non‐differentiable term in the quasistatic contact problem with the Tresca friction. The error estimates indicate that the convergence order is dependent on the nodal spacing, the time step, the largest degree of basis functions in the moving least‐squares approximation, and the penalty factor. Numerical examples demonstrate the effectiveness of the element‐free Galerkin method and verify the theoretical analysis.     

具吸收项的退化抛物问题的有限元误差估计

1　引　　言本文将考虑下列退化抛物方程的Ｇａｌｅｒｋｉｎ逼近ｕｔ =Δβ(ｕ) - ｆ(ｕ)　　在Ω× ( 0 ,Ｔ]内 ( 1 .1 )ｕ(ｘ ,ｔ) =0　　　　　　　在 Ω× ( 0 ,Ｔ]上 ( 1 .2 )ｕ(ｘ ,0 ) =ｕ0 (ｘ)　　在Ω内(1 .3)其中Ω Ｒｎ 是有界凸域 ,0 <Ｔ <∞ .β(ｖ) (ｖ∈Ｒ)是满足 β( 0 ) =β′( 0 ) =0且 β′≥ 0的函数 因此 ,( 1 1 )是退化的非线性抛物方程 方程 ( 1 1 )具有深刻的物理背景[1] ,文献 [2 - 3]讨论了方程 ( 1 1 )的特殊形式—多孔介质方程 (ＰＭＥ)的数值方法 关于ＰＭＥ解的存在性、唯一性和正则性已有许多结果 …     

Error estimates for Gaussian quadrature formulae

Summary We derive both strict and asymtotic error bounds for the Gauss-Jacobi quadrature formula with respect to a general measure. The estimates involve the maximum modulus of the integrand on a contour in the complex plane. The methods are elementary complex analysis.     

Analysis of a family of discontinuous Galerkin methods for elliptic problems: the one dimensional case

Summary. In this paper we analyze a family of discontinuous Galerkin methods, parameterized by two real parameters, for elliptic problems in one dimension. Our main results are: (1) a complete inf-sup stability analysis characterizing the parameter values yielding a stable scheme and energy norm error estimates as a direct consequence thereof, (2) an analysis of the error in L² where the standard duality argument only works for special parameter values yielding a symmetric bilinear form and different orders of convergence are obtained for odd and even order polynomials in the nonsymmetric case. The analysis is consistent with numerical results and similar behavior is observed in two dimensions.Mathematics Subject Classification (2000): 65M60, 65M15Research supported by: The Swedish Foundation for International Cooperation in Research and Higher Education     

The majorant method in the theory of newton-kantorovich approximations and the pták error estimates

For a certain modified Newton-Kontorovich method, sharp error estimates are obtained by menas of the majorant method. In particular, these error estimates generalize Pták's estimates for the usual Newton-Kontorovich method.     

Pointwise error estimates for a streamline diffusion finite element scheme

Summary Pointwise error estimates for a streamline diffusion scheme for solving a model convection-dominated singularly perturbed convection-diffusion problem are given. These estimates improve pointwise error estimates obtained by Johnson et al.[5].     

Superconvergence for a mixed finite elmenent method for elastic wave propagation in a plane domain

Summary A higher order mixed finite element method is introduced to approximate the solution of wave propagation in a plane elastic medium. A quasi-projection analysis is given to obtain error estimates in Sobolev spaces of nonpositive index. Estimates are given for difference quotients for a spatially periodic problem and superconvergence results of the same type as those of Bramble and Schatz for Galerkin methods are derived.     

The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains

In this contribution we analyze a generalization of the heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains. The method was originally introduced by E and Engquist (Commun Math Sci 1(1):87–132, 2003) for homogenization problems in fixed domains. It is based on a standard finite element approach on the macroscale, where the stiffness matrix is computed by solving local cell problems on the microscale. A-posteriori error estimates are derived in L ²(Ω) by reformulating the problem into a discrete two-scale formulation (see also, Ohlberger in Multiscale Model Simul 4(1):88–114, 2005) and using duality methods afterwards. Numerical experiments are given in order to numerically evaluate the efficiency of the error estimate.     

Spline collocation for singular integro-differential equations over (0.1)

Summary This paper analyses the convergence of spline collocation methods for singular integro-differential equations over the interval (0.1). As trial functions we utilize smooth polynomial splines the degree of which coincides with the order of the equation. Depending on the choice of collocation points we obtain sufficient and even necessary conditions for the convergence in sobolev norms. We give asymptotic error estimates and some numerical results.     

Backward error analysis for multistep methods

Summary. In recent years, much insight into the numerical solution of ordinary differential equations by one-step methods has been obtained with a backward error analysis. It allows one to explain interesting phenomena such as the almost conservation of energy, the linear error growth in Hamiltonian systems, and the existence of periodic solutions and invariant tori. In the present article, the formal backward error analysis as well as rigorous, exponentially small error estimates are extended to multistep methods. A further extension to partitioned multistep methods is outlined, and numerical illustrations of the theoretical results are presented. Received January 20, 1998 / Revised version received November 20, 1998 / Published online September 24, 1999     

Analysis of finite element methods for second order boundary value problems using mesh dependent norms

This paper presents a new approach to the analysis of finite element methods based onC ⁰-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL ₂ norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL ₂ norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.     

Complexification of Multilinear Mappings and Polynomials

We study various methods of complexifying real normed spaces. We see how the notions of duality and complexification are interchangeable. We obtain estimates for the norms of complexified multilinear mappings and polynomials. We see how polynomials can be complexified without reference to the associated multilinear mappings.