On finite element approximations to time-dependent problems

Best possible error estimates are proved for spline semi-discrete approximations to dissipative initial value problems. Error bounds are also established for suitable difference quotients.This work was supported in part by the Office of Naval Research.     

Finite element approximationof a nonlinear elliptic equation arising from bimaterial problemsin elastic-plastic mechanics

Summary. In [13], a nonlinear elliptic equation arising from elastic-plastic mechanics is studied. A well-posed weak formulation is established for the equation and some regularity results are further obtained for the solution of the boundary problem. In this work, the finite element approximation of this boundary problem is examined in the framework of [13]. Some error bounds for this approximation are initially established in an energy type quasi-norm, which naturally arises in degenerate problems of this type and proves very useful in deriving sharper error bounds for the finite element approximation of such problems. For sufficiently regular solutions optimal error bounds are then obtained for some fully degenerate cases in energy type norms. Received June 12, 1998 / Revised version received June 21, 1999 / Published online June 8, 2000     

Error Bounds for the Solution Sets of Quadratic Complementarity Problems

In this article, two types of fractional local error bounds for quadratic complementarity problems are established, one is based on the natural residual function and the other on the standard violation measure of the polynomial equalities and inequalities. These fractional local error bounds are given with explicit exponents. A fractional local error bound with an explicit exponent via the natural residual function is new in the tensor/polynomial complementarity problems literature. The other fractional local error bounds take into account the sparsity structures, from both the algebraic and the geometric perspectives, of the third-order tensor in a quadratic complementarity problem. They also have explicit exponents, which improve the literature significantly.     

Parameter-uniform numerical methods for some linear and nonlinear singularly perturbed convection diffusion boundary turning point problems

Both linear and nonlinear singularly perturbed two point boundary value problems are examined in this paper. In both cases, the problems have a boundary turning point and are of convection-diffusion type. Parameter-uniform numerical methods composed of monotone finite difference operators and piecewise-uniform Shishkin meshes, are constructed and analyzed for both the linear and the nonlinear class of problems. Numerical results are presented to illustrate the theoretical parameter-uniform error bounds established.     

ARNOLDI TYPE ALGORITHMS FOR LARGE UNSYMMETRIC MULTIPLE EIGENVALUE PROBLEMS

1.IntroductionTheLanczosalgorithm[Zo]isaverypowerfultoolforextractingafewextremeeigenvaluesandassociatedeigenvectorsoflargesymmetricmatrices[4'5'22].Sincethe1980's,considerableattentionhasbeenpaidtogeneralizingittolargeunsymmetricproblems.Oneofitsgen...     

A type of new posteriori error estimators for stokes problems

1. IntroductionIn the numerical approximation of PDE, it is often very importals to detect regionswhere the accuracy of the numerical solution is degraded by local singularities of the solutionof the continuous problem such as the singularity near the re-entrant corller. An obviousremedy is to refine the discretization in the critical regions, i.e., to place more gridpointswhere the solution is less regular. The question is how to identify these regions automdticallyand how to determine a goo…     

A parameter-uniform numerical method for a singularly perturbed two parameter elliptic problem

In this paper, a class of singularly perturbed elliptic partial differential equations posed on a rectangular domain is studied. The differential equation contains two singular perturbation parameters. The solutions of these singularly perturbed problems are decomposed into a sum of regular, boundary layer and corner layer components. Parameter-explicit bounds on the derivatives of each of these components are derived. A numerical algorithm based on an upwind finite difference operator and a tensor product of piecewise-uniform Shishkin meshes is analysed. Parameter-uniform asymptotic error bounds for the numerical approximations are established.     

On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations

Summary. This paper studies a numerical method for second-order oscillatory differential equations in which high-frequency oscillations are generated by a linear time- and/or solution-dependent part. For constant linear part, it is known that the method allows second-order error bounds independent of the product of the step-size with the frequencies and is therefore a long-time-step method. Most real-world problems are not of that kind and it is important to study more general equations. The analysis in this paper shows that one obtains second-order error bounds even in the case of a time- and/or solution-dependent linear part if the matrix is evaluated at averaged positions.Mathematics Subject Classification (2000): 65L05, 65L70Acknowledgement I am grateful to Marlis Hochbruck and Christian Lubich for helpful discussions on the subject.     

One Approach to the Comparison of Error Bounds at a Point and on a Set in the Solution of Ill-Posed Problems

The approximate solution of ill-posed problems by the regularization method always involves the issue of estimating the error. It is a common practice to use uniform bounds on the whole class of well-posedness in terms of the modulus of continuity of the inverse operator on this class. Local error bounds, which are also called error bounds at a point, have been studied much less. Since the solution of a real-life ill-posed problem is unique, an error bound obtained on the whole class of well-posedness roughens to a great extent the true error bound. In the present paper, we study the difference between error bounds on the class of well-posedness and error bounds at a point for a special class of ill-posed problems. Assuming that the exact solution is a piecewise smooth function, we prove that an error bound at a point is infinitely smaller than the exact bound on the class of well-posedness.     

On error bounds and Newton-type methods for generalized Nash equilibrium problems

Error bounds (estimates for the distance to the solution set of a given problem) are key to analyzing convergence rates of computational methods for solving the problem in question, or sometimes even to justifying convergence itself. That said, for the generalized Nash equilibrium problems (GNEP), the theory of error bounds had not been developed in depth comparable to the fields of optimization and variational problems. In this paper, we provide a systematic approach which should be useful for verifying error bounds for both specific instances of GNEPs and for classes of GNEPs. These error bounds for GNEPs are based on more general results for constraints that involve complementarity relations and cover those (few) GNEP error bounds that existed previously, and go beyond. In addition, they readily imply a Lipschitzian stability result for solutions of GNEPs, a subject where again very little had been known. As a specific application of error bounds, we discuss Newtonian methods for solving GNEPs. While we do not propose any significantly new methods in this respect, some new insights into applicability to GNEPs of various approaches and into their convergence properties are presented.     

Improved componentwise verified error bounds for least squares problems and underdetermined linear systems

Recently Miyajima presented algorithms to compute componentwise verified error bounds for the solution of full-rank least squares problems and underdetermined linear systems. In this paper we derive simpler and improved componentwise error bounds which are based on equalities for the error of a given approximate solution. Equalities are not improvable, and the expressions are formulated in a way that direct evaluation yields componentwise and rigorous estimates of good quality. The computed bounds are correct in a mathematical sense covering all sources of errors, in particular rounding errors. Numerical results show a gain in accuracy compared to previous results.     

New error bounds for the linear complementarity problem with an SB-matrix

Error bounds for SB-matrices linear complementarity problems are given in the paper (Dai et al., Numer Algorithms 61:121–139, 2012). In this paper, new error bounds for the linear complementarity problem when the matrix involved is an SB-matrix are presented and some sufficient conditions that new bounds are sharper than those of the previous paper under certain assumptions are provided. New perturbation bounds of SB-matrices linear complementarity problems are also considered.     

Upper norm bounds for the inverse of locally doubly strictly diagonally dominant matrices with its applications in linear complementarity problems

Numerical Algorithms - In this paper, we present two error bounds for the linear complementarity problems (LCPs) of locally doubly strictly diagonally dominant (LDSDD) matrices. The error bounds...     

Error bounds for stochastic shortest path problems

For stochastic shortest path problems, error bounds for value iteration due to Bertsekas elegantly generalize the classic MacQueen–Porteus error bounds for discounted infinite-horizon Markov decision problems, but incur prohibitive computational overhead. We derive bounds on these error bounds that can be computed with little or no overhead, making them useful in practice—especially so, since easily-computed error bounds have not previously been available for this class of problems.     

Vector Variational Inequalities Involving Set-valued Mappings via Scalarization with Applications to Error Bounds for Gap Functions

In this paper, by using the scalarization approach of Konnov, several kinds of strong and weak scalar variational inequalities (SVI and WVI) are introduced for studying strong and weak vector variational inequalities (SVVI and WVVI) with set-valued mappings, and their gap functions are suggested. The equivalence among SVVI, WVVI, SVI, WVI is then established under suitable conditions and the relations among their gap functions are analyzed. These results are finally applied to the error bounds for gap functions. Some existence theorems of global error bounds for gap functions are obtained under strong monotonicity and several characterizations of global (respectively local) error bounds for the gap functions are derived.     

一维有限元的h-p方法的后验误差估计

有限元的h-p方法,是指在增加有限元空间的维数时,既加密某些单元的网格,同时也增加某些单元的次数.对h-p方法,人们希望得到O(h~mp~(-n))(m,n>0)形状的误差估计.这种误差估计的结果包括了对传统的h方法以及p方法的结果.关于h-p方法的     

Error bounds of two smoothing approximations for semi-infinite minimax problems

In the paper we investigate smoothing method for solving semi-infinite minimax problems. Not like most of the literature in semi-infinite minimax problems which are concerned with the continuous time version（i.e., the one dimensional semi-infinite minimax problems）, the primary focus of this paper is on multi- dimensional semi-infinite minimax problems. The global error bounds of two smoothing approximations for the objective function are given and compared. It is proved that the smoothing approximation given in this paper can provide a better error bound than the existing one in literature.     

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.     

Approximate Solutions and Error Bounds for Solving Matrix Differential Equations Without Increasing the Dimension of the Problem

In this paper we present a method for solving initial value problems related to second order matrix differential equations. This method is based on the existence of a solution of a certain algebraic matrix equation related to the problem, and it avoids the increase of the dimension of the problem for its resolution. Approximate solutions, and their error bounds in terms of error bounds for the approximate solutions of the algebraic problem, are given.     

Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems

The paper is devoted to the problem of verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered in this work. Based on functional type estimates developed on an abstract level, we present a general technology for constructing computable sharp upper bounds for the global error for various particular classes of elliptic problems. Here the global error is understood as a suitable energy type difference between the true and computed solutions. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions, and are sharp in the sense that they can be, in principle, made as close to the true error as resources of the used computer allow. The latter can be achieved by suitably tuning the auxiliary parameter functions, involved in the proposed upper error bounds, in the course of the calculations.