Bounds for aggregating nodes in network problems

It is often necessary or desirable in practice to replace a large, detailed optimization model with a smaller, approximate model. It would be useful to have bounds on the error resulting from this process of aggregation. This paper develops such bounds for a class of linear minimum-cost network-flow problems, where groups of nodes in a large problem are replaced by aggregate nodes. Two types of bounds are derived: A priori bounds are available before solving the aggregated problem; a posteriori bounds, which are generally tighter, may be computed afterwards.     

Quasi-norm a priori and a posteriori error estimates for the nonconforming approximation of p-Laplacian

Summary. In this paper, we derive quasi-norm a priori and a posteriori error estimates for the Crouzeix-Raviart type finite element approximation of the p-Laplacian. Sharper a priori upper error bounds are obtained. For instance, for sufficiently regular solutions we prove optimal a priori error bounds on the discretization error in an energy norm when . We also show that the new a posteriori error estimates provide improved upper and lower bounds on the discretization error. For sufficiently regular solutions, the a posteriori error estimates are further shown to be equivalent on the discretization error in a quasi-norm. Received January 25, 1999 / Revised version received June 5, 2000 Published online March 20, 2001     

A discontinuous Galerkin method for the Rosenau equation

A priori error estimates for the Rosenau equation, which is a K-dV like Rosenau equation modelled to describe the dynamics of dense discrete systems, have been studied by one of the authors. But since a priori error bounds contain the unknown solution and its derivatives, it is not effective to control error bounds with only a given step size. Thus we need to estimate a posteriori errors in order to control accuracy of approximate solutions using variable step sizes. A posteriori error estimates of the Rosenau equation are obtained by a discontinuous Galerkin method and the stability analysis is discussed for the dual problem. Numerical results on a posteriori error and wave propagation are given, which are obtained by using various spatial and temporal meshes controlled automatically by a posteriori error.     

A posteriori error analysis of the discontinuous finite element methods for first order hyperbolic problems

In this paper, we consider the a posteriori error analysis of discontinuous Galerkin finite element methods for the steady and nonsteady first order hyperbolic problems with inflow boundary conditions. We establish several residual-based a posteriori error estimators which provide global upper bounds and a local lower bound on the error. Further, for nonsteady problem, we construct a fully discrete discontinuous finite element scheme and derive the a posteriori error estimators which yield global upper bound on the error in time and space. Our a posteriori error analysis is based on the mesh-dependent a priori estimates for the first order hyperbolic problems. These a posteriori error analysis results can be applied to develop the adaptive discontinuous finite element methods.     

Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications

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.     

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.     

Guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem

We derive guaranteed a posteriori error estimates for nonconforming finite element approximations to a singularly perturbed reaction–diffusion problem. First, an abstract a posteriori error bound is derived under a special equilibration condition. Based on conservative flux reconstruction, two error estimators are proposed and provide actual upper error bounds in the usual energy norm without unknown constants, one of which can be directly constructed without solving local Neumann problems and provide practical computable error bounds. The error estimators also provide local lower bounds but with the multiplicative constants dependent on the diffusion coefficient and mesh size, where the constants can be bounded for enough small mesh size comparable with the square root of the diffusion coefficient. By adding edge jumps with weights to the energy norm, two modified error estimators with additional edge tangential jumps are shown to be robust with respect to the diffusion coefficient and provide guaranteed upper bounds on the error in the modified norm. Finally, the performance of the estimators are illustrated by the numerical results.     

Residual type a posteriori error estimates for elliptic obstacle problems

Summary. A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positivity. Both upper and lower bounds are proved and their optimality is explored with several examples. Sharp a priori bounds for the a posteriori estimators are given, and extensions of the results to double obstacle problems are briefly discussed. Received June 19, 1998 / Published online December 6, 1999     

Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem

A non-monotone FEM discretization of a singularly perturbed one-dimensional reaction-diffusion problem whose solution exhibits strong layers is considered. The method is shown to be maximum-norm stable although it is not inverse monotone. Both a priori and a posteriori error bounds in the maximum norm are derived. The a priori result can be used to deduce uniform convergence of various layer-adapted meshes proposed in the literature. Numerical experiments complement the theoretical results. AMS subject classification (2000)  65L10, 65L50, 65L60     

A posteriori and constructive a priori error bounds for finite element solutions of the Stokes equations

We describe a method to estimate the guaranteed error bounds of the finite element solutions for the Stokes problem in mathematically rigorous sense. We show that an a posteriori error can be computed by using the numerical estimates of a constant related to the so-called inf-sup condition for the continuous problem. Also a method to derive the constructive a priori error bounds are considered. Some numerical examples which confirm us the expected rate of convergence are presented.     

Variational approximations in energy space for the regulator problem with time delay

We consider a variational procedure for approximating the solution of the state regulator problem with time delay. Motivated by a dual formulation of the problem, we introduce a positive-definite functionalF over a certain energy space of Mikhlin and obtain approximating solutions by the Ritz-Trefftz idea of minimizing it over finite-dimensional subspaces. The resulting approximating solutions, in turn, furnish suboptimal solutions which converge to the optimal solution of the regulator problem with time delay. A priori error bounds in terms of splines are given. A posteriori error bounds are also obtained.     

A priori and a posteriori error analyses in the study of viscoelastic problems

In this work, the numerical approximation of a viscoelastic problem is studied. A fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize time derivatives. Then, two numerical analyses are presented. First, a priori estimates are proved from which the linear convergence of the algorithm is derived under suitable regularity conditions. Secondly, an a posteriori error analysis is provided extending some preliminary results obtained in the study of the heat equation. Upper and lower error bounds are obtained.     

Defect-based local error estimators for splitting methods,with application to Schrödinger equations,Part I: The linear case

We introduce a defect correction principle for exponential operator splitting methods applied to time-dependent linear Schrödinger equations and construct a posteriori local error estimators for the Lie–Trotter and Strang splitting methods. Under natural commutator bounds on the involved operators we prove asymptotical correctness of the local error estimators, and along the way recover the known a priori convergence bounds. Numerical examples illustrate the theoretical local and global error estimates.     

Numerical Analysis of a Nonlinear Singularly Perturbed Delay Volterra Integro-Differential Equation on an Adaptive Grid

In this paper, we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.     

A posteriori error analysis for the normal compliance problem

In this work, a contact problem between a linear elastic material and a deformable obstacle is numerically analyzed. The contact is modeled using the well-known normal compliance contact condition. The weak formulation leads to a nonlinear variational equation which is approximated by using the finite element method. A priori error estimates are recalled. Then, we define an a posteriori error estimator of residual type to evaluate the accuracy of the finite element approximation of the problem. Upper and lower bounds of the discretization error are proved for this estimator.     

Stabilized finite element method for Navier-Stokes equations with physical boundary conditions

This paper deals with the numerical approximation of the 2D and 3D Navier-Stokes equations, satisfying nonstandard boundary conditions. This lays on the finite element discretisation of the corresponding Stokes problem, which is achieved through a three-fields stabilized mixed formulation. A priori and a posteriori error bounds are established for the nonlinear problem, ascertaining the convergence of the method. Finally, numerical tests are presented, including mesh refinement via error indicators.

     

Identifying an unknown source in the Poisson equation by the method of Tikhonov regularization in Hilbert scales

In this paper, we consider the problem for identifying the unknown source in the Poisson equation. The Tikhonov regularization method in Hilbert scales is extended to deal with illposedness of the problem and error estimates are obtained with an a priori strategy and an a posteriori choice rule to find the regularization parameter. The user does not need to estimate the smoothness parameter and the a priori bound of the exact solution when the a posteriori choice rule is used. Numerical examples show that the proposed method is effective and stable.     

A posteriori error estimates for the Crank-Nicolson method for parabolic equations

We derive optimal order a posteriori error estimates for time discretizations by both the Crank-Nicolson and the Crank-Nicolson-Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are second-order Crank-Nicolson reconstructions of the piecewise linear approximate solutions. These functions satisfy two fundamental properties: (i) they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and (ii) they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank-Nicolson method.

     

A normal compliance contact problem in viscoelasticity: An a posteriori error analysis and computational experiments

In this work, the numerical approximation of a viscoelastic contact problem is studied. The classical Kelvin-Voigt constitutive law is employed, and contact is assumed with a deformable obstacle and modelled using the normal compliance condition. The variational formulation leads to a nonlinear parabolic variational equation. An existence and uniqueness result is recalled. Then, a fully discrete scheme is introduced, by using the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize time derivatives. A priori error estimates recently proved for this problem are recalled. Then, an a posteriori error analysis is provided, extending some preliminary results obtained in the study of the heat equation and other parabolic equations. Upper and lower error bounds are proved. Finally, some numerical experiments are presented to demonstrate the accuracy and the numerical behaviour of the error estimates.     

A new a posteriori error estimate for the Morley element

In this paper, we present a posteriori error analysis for the nonconforming Morley element of the fourth order elliptic equation. We propose a new residual-based a posteriori error estimator and prove its reliability and efficiency. These results refine those of Beirao da Veiga et al. (Numer Math 106:165–179, 2007) by dropping two edge jump terms in both the energy norm of the error and the estimator, and those of Wang and Zhang (Local a priori and a posteriori error estimates of finite elements for biharmonic equation, Research Report, 13, 2006) by showing the efficiency in the sense of Verfürth (A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley-Teubner, New York, 1996). Moreover, the normal component in the estimators of Beirao da Veiga et al. (Numer Math 106:165–179, 2007) and Wang and Zhang (Local a priori and a posteriori error estimates of finite elements for biharmonic equation, Research Report, 13, 2006) is dropped, and therefore only the tangential component of the stress on each edge comes into the estimator. In addition, we generalize these results to three dimensional case.