Runge-Kutta Methods with Global Error Estimates

In the numerical solution of ordinary differential initial valueproblems, it is desirable to have a global error estimate. Inthis paper the practical use of a global error estimate introducedby H. J. Stetter is discussed, and specific procedures involvingexplicit Runge-Kutta methods, which effect a certain savingin function evaluations, are presented. Some numerical resultsshow the accuracy of the estimate.     

A-Posteriori Local Error Estimates of Boundary Element Methods with Some Pseudo-Differential Equations on Closed Curves

In this paper we show local error estimates for the Galerkin finite element method applied to strongly elliptic pseudo-differential equations on closed curves. In these local estimates the right hand sides are obtained as the sum of a local norm of the residual, which is computable, and additional terms of higher order with respect to the computable, and additional terms of higher order with respect to the meshwidth. Hence, asymptotically, here the residual is an error indicator which provides a corresponding self-adaptive boundary element method.     

Error Estimates for Mixed Finite Element Methods for Sobolev Equation

1 IntroductionLet fl be a bounded domain in R2 with Lipschitz continuous boundaxy 0fl. For thed0 < T < co, we consider the fo1lowing initial-boun'lar}-ralue problem for thc Sobolevequation:where ut denotes the time derivative of the function (1. Vu denotes the gradient of thefunction u, and divv denotes the divergence of the vect{Jr tulued function v, a1 b1, f, anduo are known functions.The standard finite element method for (1.1) (1.3) llas received considerable attentionand is well studied…     

Global Convergence Analysis of Line Search Interior-Point Methods for Nonlinear Programming without Regularity Assumptions

As noted by Wächter and Biegler (Ref. 1), a number of interior-point methods for nonlinear programming based on line-search strategy may generate a sequence converging to an infeasible point. We show that, by adopting a suitable merit function, a modified primal-dual equation, and a proper line-search procedure, a class of interior-point methods of line-search type will generate a sequence such that either all the limit points of the sequence are KKT points, or one of the limit points is a Fritz John point, or one of the limit points is an infeasible point that is a stationary point minimizing a function measuring the extent of violation to the constraint system. The analysis does not depend on the regularity assumptions on the problem. Instead, it uses a set of satisfiable conditions on the algorithm implementation to derive the desired convergence property.Communicated by Z. Q. LuoThis research was partially supported by Grant R-314-000-026/042/057-112 of National University of Singapore and Singapore-MIT Alliance. We thank Professor Khoo Boo Cheong, Cochair of the High Performance Computation Program of Singapore-MIT Alliance, for his support     

Error Estimates for Three Methods of Evaluating Cauchy Principal Value Integrals

In this paper, we give error expressions for the subtractionof the singularity method, the method of symmetric pairing anda method of L. M. Delves in the evaluation of the PrincipalValue of the improper integral The error form for the subtraction of the singularity methodis used to deduce the important property that the numericalmethod for the value of the improper integral remains stableas the value of x approaches either of the end-point valuesa and b.     

球面上径向基函数最小范数插值逼近的误差估计

研究了球面径向基插值对球面函数的逼近问题,给出了一致逼近的上界估计式．文中结果说明,球面径向基插值的逼近阶会随函数光滑性的提高而增加．     

On the Randomized Error of Polynomial Methods for Eigenvector and Eigenvalue Estimates

In this paper we consider the problem of estimating the largest eigenvalue and the corresponding eigenvector of a symmetric matrix. In particular, we consider iterative methods, such as the power method and the Lanczos method. These methods need a starting vector which is usually chosen randomly. We analyze the behavior of these methods when the initial vector is chosen with uniform distribution over the unitn-dimensional sphere. We extend and generalize the results reported earlier. In particular, we give upper and lower bounds on the _pnorm of the randomized error, and we improve previously known bounds with a detailed analysis of the role of the multiplicity of the largest eigenvalue.     

A Posteriori Error Estimates of Mixed Methods for Parabolic Optimal Control Problems

In this article, we shall give a brief review on the fully discrete mixed finite element method for general optimal control problems governed by parabolic equations. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. Furthermore, we derive a posteriori error estimates for the finite element approximation solutions of optimal control problems. Some numerical examples are given to demonstrate our theoretical results.     

非线性抛物型方程组的二次有限体积元方法及其误差估计

讨论基于三角形网格的二维非线性抛物型方程组的有限体积元方法,其中试探函数空间为二次Lagrange元,检验函数空间为分片常数函数空间,对问题的全离散格式证明了最优的能量模误差估计。最后给出一个相关数值算例以验证格式的有效性。     

Error Estimates for the Finite Element Solutions of Some Variational Inequalities with Nonlinear Monotone Operator

In this paper, we estimate the error of the linear finite element solutions of the obstacle problem and the unilateral problem with monotone operator. We obtained $O(h)$ error bound for the obstacle problem and $O(h^{3/4})$ error bound for the unilateral problem. And if the solution $u^*$ of the unilateral problem possesses more smoothness, then $O(h)$ error bound can be obtained in the same way as [2].     

Strichartz Estimates for Wave Equations with Coefficients of Sobolev Regularity

Wave packet techniques provide an effective method for proving Strichartz estimates on solutions to wave equations whose coefficients are not smooth. We use such methods to show that the existing results for C ^{1, 1} and C ^{1, α} coefficients can be improved when the coefficients of the wave operator lie in a Sobolev space of sufficiently high order.     

Some New Extrapolation Estimates for the Scale of Spaces

Using new extrapolation estimates for the - and -functionals of couples of limit spaces of the -scale , we introduce a class of extrapolation functors. A characterization of this class via the real interpolation method permits one to obtain new equivalent expressions for the norms in symmetric spaces close to and , which depend only on the -norms of a function.     

粘弹性波动方程的H~1-Galerkin混合有限元方法误差估计(英文)

H1-Galerkin mixed methods are proposed for viscoelasticity wave equation.Depending on the physical quantities of interest,two methods are discussed.The optimal error estimates and the proof of the existence and uniqueness of semidiscrete solutions are derived for problems in one space dimension.And the methods don＇t require the LBB condition.     

Boundary Regularity for Minimal Graphs of Higher Codimensions

In this paper, the authors derive H¨older gradient estimates for graphic functions of minimal graphs of arbitrary codimensions over bounded open sets of Euclidean space under some suitable conditions.     

Numerical Methods and Error Estimates of Nonlinear Tw''O-Point Boundary Value Problem with Discontinous Coefficients

In this paper we consider L^\infty error estimates for difference methods and Galerkin methods to solutions of nonlinear two-point boundary value problem $[\left\{ {\begin{array}{*{20}{c}} { - D(pDu) + qu = f(x,u),0 < x < 1,}\{u(0) = u(1) = 0,} \end{array}} \right.\]$ Where coefficient function p(x) is pieoewise continuous. Moreover, discontinuous points are not chosen to be a mesh points.