On the convergence rate of vanishing viscosity approximations

Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound ‖u(t, ·) ? u^?(t, ·)‖ = O(1)(1 + t) · |ln ?| on the distance between an exact BV solution u and a viscous approximation u^?, letting the viscosity coefficient ? → 0. In the proof, starting from u we construct an approximation of the viscous solution u^? by taking a mollification u * and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed ?. Error estimates are then obtained by introducing new Lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families. © 2004 Wiley Periodicals, Inc.     

On the best approximations and rate of convergence of decompositions in the root vectors of an operator

We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator A that acts in B. The corresponding estimates of the best approximations are expressed in terms of a K-functional associated with the operator A. For the operator of differentiation with periodic boundary conditions, these estimates coincide with the classical Jackson inequalities for the best approximations of functions by trigonometric polynomials. In terms of K-functionals, we also prove the abstract Dini-Lipschitz criterion of convergence of partial sums of the decomposition of f from B in the root vectors of the operator A to f     

On sufficient conditions for the convergence of successive approximations

We obtain sufficient conditions for the convergence of successive approximations to the solution of operator equations. We show how these conditions can be used in the case of nonlinear integral Hammerstein-type equations. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 854–859, June, 1999.     

On the accuracy and convergence of conjugate flux approximations

We prove that the L²-projections of derivatives of piecewise bilinear or linear finite element approximations of smooth solutions of elliptic boundary-value problems on the interior of uniform meshes, converge in L² and L^∞ at a rate faster than that of derivatives of the approximations themselves.     

On optimal convergence for regularized Tikhonov approximations

We consider the generalized solution to a Fredholm integral equation of the first kind, and establish conditions for which the Tikhonov and Raleigh-Ritz approximations converge to the true solution at the same rate as the perturbed data approaches to the exact data.     

Optimal convergence rate for Yosida approximations of bounded holomorphic semigroups

In this paper, we obtain optimal bounds for convergence rate for Yosida approximations of bounded holomorphic semigroups. We also provide asymptotic expansions for semigroups in terms of Yosida approximations and obtain optimal error bounds for these expansions.     

On the convergence of the multigroup approximations for multidimensional media

Summary In this paper, we investigate the stability, convergence, and consistency properties of steady-state multigroup models for submultiplying media with spatial dimensions greater than one. We define these concepts in a Banach space whose norm measures the collision density integrated over phase space. Stability and consistency occur under the conditions that the maximum fluctuations in the total cross section, and in the expected number of secondary particles arising from each energy level, tend to zero as the energy mesh becomes finer. A concluding example and discussion deal with pathologies of the multigroup model in situations where these fluctuations do not tend to zero as the norms of the energy partitions. The results in this paper complement the time-dependent results of both Belleni-Morante and Busoni for isotropic slabs and of Yang Mingzhu and Zhu Guangtian for bounded, three-dimensional media. This work directly extends the steady-state results of Paul Nelson, Jr. and H. D. Victory, Jr. for slab media to steady-state transport in multidimensional media.This research was partially supported by the Alexander von Humboldt Foundation and by a Faculty Development Leave from Texas Tech University during the academic year 1982–1983. Partial support was also provided in the initial stages of this work by the U.S. National Science Foundation under Grant CPE 8007396.     

Estimates of rate of convergence of best approximations by local functions

Questions are considered on the rate of convergence (in some abstract space of functions) of approximations that are the best in another space. Under specific conditions it is shown that the best approximations by local functions in a weighted Sobolev space W _p,B ^r yield almost-best approximation in W _q,B ^r with q[p,+).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 22–37, 1976.     

On rates of convergence for approximations to the generalized inverse

A rate of convergence result for a general class of approximation methods for the generalized inverse of a bounded linear operator with arbitrary range is presented. Applications are given to a number of iterative and noniterative approximation techniques.     

The rate of convergence of weak greedy approximations over orthogonal dictionaries

The convergence rate of a weak orthogonal greedy algorithm is studied for the subspace ?¹ ? ?² and orthogonal dictionaries. It is shown that general results on convergence rate of weak orthogonal greedy algorithms can be essentially improved in the studied case. It is also shown that this improvement is asymptotically sharp.     

On the speed of convergence in the method of Fejer approximations

The problem of the speed of convergence in the method of Fejer approximations is examined, with an application to the problem of finding at least one solution of the system of inequalitiesf _j(x) 0, j=1,..., m, where thef_j(x) are smooth convex functions defined on a real Hilbert space.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 53–62, July, 1968.     

On the convergence of the multigroup approximations for submultiplying slab media

The stability, convergence, and consistency properties of the steady-state multigroup model are investigated for submultiplying slab media. These concepts are defined in a Banach space setting in which the norm of the angular flux is the collision density integrated over phase space. It is shown that the multigroup approximations are stable and are both consistent with, and convergent to, the transport equation under the conditions that the maximum fluctuations in the total cross section and in the expected number of secondary particles, arising from each energy level, tend to zero as the energy mesh becomes finer. A concluding discussion deals with pathologies of the multigroup approximation for situations in which these fluctuations do not tend to zero as the norms of the energy partitions. The results in this paper complement the time-dependent results of Belleni-Morante and Busoni for isotropic slabs and the results of Nelson for steady-state rod media.     

On the convergence of rational approximations of semigroups on intermediate spaces

We generalize a result by Brenner and Thomée on the rate of convergence of rational approximation schemes for semigroups. Using abstract interpolation techniques we obtain convergence on a continuum of intermediate spaces between the Banach space and the domain of a certain power of the generator of the semigroup. The sharpness of the results is also discussed.

     

On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems

Consider a linear ordinary differential equation of the 2nd order which has a singularity at the origin; according to the nature of this singularity we must consider either the two-point boundary-value problem or the one-point boundary value problem. Finite-difference schemes are studied; results are given concerning error analysis and monotone convergence.The material of this paper is taken from the author's doctoral thesis [2]. The author is greatly indebted to his adviser S. V. Parter. - Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No.: DA-31-124-ARO-D-462.     

On the convergence rate of Fourier series

The rate of convergence of a Fourier‐series representation of a given function depends on the nature of the function and of its derivatives. This is shown by using the graphical outputs of a desk computer for different cases. For full‐range series, the effects of continuity and discontinuity of the function and its first derivatives are shown first. The advantage of half‐range formulae due to the free choice of function in the second half‐range are demonstrated next, along with the importance of choice of sine and cosine series according to the function being represented. Finally, a method is given of modifying the given function in a simple manner whereby a dramatic increase in the rate of convergence of the Fourier series is obtained. Examples of the Gibbs overshoot and its elimination are included.