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1.
In this paper, we introduce the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities.
Elliptic quadratic inequalities are closely related to Chebyshev approximation of vector-valued functions (including complex-valued
functions). The set of Chebyshev approximations of a vector-valued function defined on a finite set is shown to be Hausdorff
strongly unique of order exactly 2
s
for some nonnegative integer s. As a consequence, the exact order of Hoffman’s error bounds for approximate solutions of elliptic quadratic inequalities
is exactly 2
-s
for some nonnegative integer s. The integer s, called the order of deficiency (which is computable), quantifies how much the Abadie constraint qualification is violated
by the elliptic quadratic inequalities.
Received: April 15, 1999 / Accepted: February 21, 2000?Published online July 20, 2000 相似文献
2.
Summary. We prove convergence results and error estimates for interpolatory product quadrature formulas for Cauchy principal value
integrals on the real line with Freud–type weight functions. The formulas are based on polynomial interpolation at the zeros
of orthogonal polynomials associated with the weight function under consideration. As a by–product, we obtain new bounds for
the derivative of the functions of the second kind for these weight functions.
Received July 15, 1997 / Revised version received August 25, 1998 相似文献
3.
Summary. In this paper we design high-order local artificial boundary conditions and present error bounds for the finite element approximation
of an incompressible elastic material in an unbounded domain. The finite element approximation is formulated in a bounded
computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family
of nonlocal approximate artificial boundary conditions with increasing accuracy (and computational cost) and a family of local
ones for a given artificial boundary. Our error bounds indicate how the errors of the finite element approximations depend
on the mesh size, the terms used in the approximate artificial boundary condition and the location of the artificial boundary.
Numerical examples of an incompressible elastic material outside a circle in the plane is presented. Numerical results demonstrate
the performance of our error bounds.
Received August 31, 1998 / Revised version received November 6, 2001 / Published online March 8, 2002 相似文献
4.
Summary. We construct a new algorithm for the numerical integration of functions that are defined on a -dimensional cube. It is based on the Clenshaw-Curtis rule for and on Smolyak's construction. This way we make the best use of the smoothness properties of any (nonperiodic) function.
We prove error bounds showing that our algorithm is almost optimal (up to logarithmic factors) for different classes of functions
with bounded mixed derivative. Numerical results show that the new method is very competitive, in particular for smooth integrands
and .
Received April 3, 1995 / Revised version received November 27, 1995 相似文献
5.
Shayne Waldron 《Numerische Mathematik》2000,85(3):469-484
Summary. Distribution theory is used to construct minimally supported Peano kernel type representations for linear functionals such as the error in multivariate Hermite interpolation. The simplest
case is that of representing the error in approximation to f by the constant polynomial f(a) in terms of integrals of the first order derivatives of f. This is discussed in detail. Here it is shown that suprisingly there exist many representations which are not minimally
supported, and involve the integration of first order derivatives over multidimensional regions. The distance of smooth functions
from the constants in the uniform norm is estimated using our representations for the error.
Received June 30, 1997 / Revised version received April 6, 1999 / Published online February 17, 2000 相似文献
6.
Kai Diethelm 《Numerische Mathematik》1996,73(1):53-63
Summary.
We show that, if
(),
the error term of
every modified positive interpolatory quadrature rule for
Cauchy principal value integrals of the type
,
, fulfills
uniformly for all
, and hence it is
of optimal
order of magnitude in the classes
().
Here, is a weight function with the property
.
We give explicit upper bounds for the Peano-type error
constants of such rules.
This improves and completes earlier results by
Criscuolo and Mastroianni
(Calcolo 22 (1985), 391–441 and Numer. Math.
54 (1989), 445–461)
and Ioakimidis (Math. Comp. 44 (1985), 191–198).
For the special case of the Gaussian rule, we
show that the restriction
can be dropped.
The results are based on a new representation of the
Peano kernels of these formulae via the Peano kernels of the underlying
classical quadrature formulae. This representation may also be
useful in connection with some different problems.
Received November 21, 1994 相似文献
7.
Summary. We generalize earlier results concerning
an asymptotic error expansion of wavelet
approximations. The properties of the monowavelets,
which are the building
blocks for the error expansion, are studied in more
detail, and connections
between spline wavelets and Euler and
Bernoulli polynomials are pointed out.
The expansion is used to compare the
error for different wavelet families.
We prove that the leading terms of the
expansion only depend on the multiresolution
subspaces and not
on how the complementary subspaces
are chosen.
Consequently, for a fixed set of
subspaces , the leading
terms do not depend on the fact whether
the wavelets are orthogonal or not.
We also show that Daubechies' orthogonal wavelets need,
in general, one level more than spline wavelets to obtain an
approximation with a prescribed accuracy.
These results are illustrated with numerical examples.
Received May 3, 1993 / Revised version received January 31, 1994 相似文献
8.
Recently, in the article [LW], the authors use the notion of polynomials in metric spaces of homogeneous type (in the sense of Coifman-Weiss) to prove a relationship between high order Poincaré inequalities and
representation formulas involving fractional integrals of high order, assuming only that is a doubling measure and that geodesics exist. Motivated by this and by recent work in [H], [FHK], [KS] and [FLW] about
first order Sobolev spaces in metric spaces, we define Sobolev spaces of high order in such metric spaces . We prove that several definitions are equivalent if functions of polynomial type exist. In the case of stratified groups,
where polynomials do exist, we show that our spaces are equivalent to the Sobolev spaces defined by Folland and Stein in [FS].
Our results also give some alternate definitions of Sobolev spaces in the classical Euclidean case.
Received: 10 February 1999 / Published online: 1 February 2002 相似文献
9.
Summary.
The existence of a true orbit near a numerically
computed approximate orbit -- shadowing -- of
autonomous system of ordinary differential equations
is investigated.
A general shadowing theorem for finite time,
which guarantees the existence of shadowing
in ordinary differential equations
and provides error bounds for the distance between
the true and the approximate orbit in terms of computable
quantities, is proved.
The practical use and the effectiveness of this theorem
is demonstrated in the numerical computations
of chaotic orbits of the Lorenz equations.
Received December 15, 1993 相似文献
10.
Summary.
We consider the finite element approximation of a
non-Newtonian flow, where the viscosity obeys a general law including
the Carreau or power law. For sufficiently regular solutions we prove
energy type error bounds for the velocity and pressure. These bounds
improve on existing results in the literature. A key step in the
analysis is to prove abstract error bounds initially in a quasi-norm,
which naturally arises in degenerate problems of this type.
Received May 25, 1993 / Revised version received January 11, 1994 相似文献
11.
Summary. We consider the bilinear finite element approximation of smooth solutions to a simple parameter dependent elliptic model
problem, the problem of highly anisotropic heat conduction. We show that under favorable circumstances that depend on both
the finite element mesh and on the type of boundary conditions, the effect of parametric locking of the standard FEM can be
reduced by a simple variational crime. In our analysis we split the error in two orthogonal components, the approximation
error and the consistency error, and obtain different bounds for these separate components. Also some numerical results are
shown.
Received September 6, 1999 / Revised version received March 28, 2000 / Published online April 5, 2001 相似文献
12.
Denote by the error of a Romberg quadrature rule applied to the function f. We determine approximately the constants in the bounds of the types and
for all classical Romberg rules. By a comparison with the corresponding constants of the Gaussian rule we give the statement
“The Gaussian quadrature rule is better than the Romberg method” a precise meaning.
Received September 10, 1997 / Revised version received February 16, 1998 相似文献
13.
Given , we consider the following problem: find , such that where or 3, and in . We prove and error bounds for the standard continuous piecewise linear Galerkin finite element approximation with a (weakly) acute triangulation.
Our bounds are nearly optimal. In addition, for d = 1 and 2 and we analyze a more practical scheme involving numerical integration on the nonlinear term. We obtain nearly optimal and error bounds for d = 1. For this case we also present some numerical results.
Received July 4, 1996 / Revised version received December 18, 1997 相似文献
14.
Alastair Spence 《Numerische Mathematik》1979,32(2):139-146
Summary In a previous paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of non-symmetric integral equations. In this note an alternative analysis is presented leading to equivalent dominant error terms with error bounds which are quicker to calculate than those derived previously. 相似文献
15.
On the Zero-Divergence of Equidistant Lagrange Interpolation 总被引:1,自引:0,他引:1
Michael Revers 《Monatshefte für Mathematik》2000,131(3):215-221
In 1942, P. Szász published the surprising result that if a function f is of bounded variation on [−1, 1] and continuous at 0 then the sequence of the equidistant Lagrange interpolation polynomials
converges at 0 to . In the present note we give a construction of a function continuous on [−1, 1] whose Lagrange polynomials diverge at 0.
Moreover, we show that the rate of divergence attains almost the maximal possible rate.
(Received 2 February 2000) 相似文献
16.
Summary. In this paper we analyse the existence of asymptotic expansions of the error of Galerkin methods with splines of arbitrary
degree for the approximate solution of integral equations with logarithmic kernels. These expansions are obtained in terms
of an interpolation operator and are useful for the application of Richardson extrapolation and for obtaining sharper error
bounds. We also present and analyse a family of fully discrete spline Galerkin methods for the solution of the same equations.
Following the analysis of Galerkin methods, we show the existence of asymptotic expansions of the error.
Received May 18, 1995 / Revised version received April 11, 1996 相似文献
17.
Summary. For analytic functions the remainder term of quadrature rules can be represented as a contour integral with a complex kernel
function. From this representation different remainder term estimates involving the kernel are obtained. It is studied in
detail how polynomial biorthogonal systems can be applied to derive sharp bounds for the kernel function. It is shown that
these bounds are practical to use and can easily be computed. Finally, various numerical examples are presented.
Received March 11, 1998 / Revised version January 22, 1999/ Published online November 17, 1999 相似文献
18.
Summary. We define the multivariate Padé-Bergman approximants (also called Padé approximants) and prove a natural generalization of de Montessus de Ballore theorem.
Numerous definitions of multivariate Padé approximants have already been introduced. Unfortunately, they all failed to generalize
de Montessus de Ballore theorem: either spurious singularities appeared (like the homogeneous Padé [3,4], or no general convergence
can be obtained due to the lack of consistency (like the equation lattice Padé type [3]).
Recently a new definition based on a least squares approach shows its ability to obtain the desired convergence [6]. We improve
this initial work in two directions. First, we propose to use Bergman spaces on polydiscs as a natural framework for stating
the least squares problem. This simplifies some proofs and leads us to the multivariate Padé approximants. Second, we pay a great attention to the zero-set of multivariate polynomials in order to find weaker (although
natural) hypothesis on the class of functions within the scope of our convergence theorem. For that, we use classical tools
from both algebraic geometry (Nullstellensatz) and complex analysis (analytic sets, germs).
Received December 4, 2001 / Revised version received January 2, 2002 / Published online April 17, 2002 相似文献
19.
Summary. Interpolation with translates of a basis function is a common process in approximation theory. The most elementary form of
the interpolant consists of a linear combination of all translates by interpolation points of a single basis function. Frequently,
low degree polynomials are added to the interpolant. One of the significant features of this type of interpolant is that it
is often the solution of a variational problem. In this paper we concentrate on developing a wide variety of spaces for which
a variational theory is available. For each of these spaces, we show that there is a natural choice of basis function. We
also show how the theory leads to efficient ways of calculating the interpolant and to new error estimates.
Received December 10, 1996 / Revised version received August 29, 1997 相似文献
20.
Summary. We study the -stability and error estimates of general approximate solutions for the Cauchy problem associated with multidimensional Hamilton-Jacobi
(H-J) equations. For strictly convex Hamiltonians, we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors. We then demonstrate this general theory
for two types of approximations: approximate solutions constructed by the vanishing viscosity method, and by Godunov-type
finite difference methods. If we let denote the `small scale' of such approximations (– the viscosity amplitude , the spatial grad-size , etc.), then our -error estimates are of , and are sharper than the classical -results of order one half, . The main building blocks of our theory are the notions of the semi-concave stability condition and -measure of the truncation error. The whole theory could be viewed as a multidimensional extension of the -stability theory for one-dimensional nonlinear conservation laws developed by Tadmor et. al. [34,24,25]. In addition, we
construct new Godunov-type schemes for H-J equations which consist of an exact evolution operator and a global projection operator. Here, we restrict our attention to linear projection operators (first-order schemes). We note, however,
that our convergence theory applies equally well to nonlinear projections used in the context of modern high-resolution conservation laws. We prove semi-concave stability and obtain -bounds on their associated truncation errors; -convergence of order one then follows. Second-order (central) Godunov-type schemes are also constructed. Numerical experiments
are performed; errors and orders are calculated to confirm our -theory.
Received April 20, 1998 / Revised version received November 8, 1999 / Published online August 24, 2000 相似文献