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151.
Erich Novak 《Numerische Mathematik》1986,50(2):245-252
Summary The definition of the average error of numerical methods (by example of a quadrature formula
to approximateS(f)= f d on a function classF) is difficult, because on many important setsF there is no natural probability measure in the sense of an equidistribution. We define the average a posteriori error of an approximation
by an averaging process over the set of possible information, which is used by
(in the example of a quadrature formula,N(F)={(f(a
1), ...,f/fF} is the set of posible information). This approach has the practical advantage that the averaging process is related only to finite dimensional sets and uses only the usual Lebesgue measure. As an application of the theory I consider the numerical integration of functions of the classF={f:[0,1]/f(x)–f(y)||x–y|}. For arbitrary (fixed) knotsa
i
we determine the optimal coefficientsc
i
for the approximation
and compute the resulting average error. The latter is minimal for the knots
. (It is well known that the maximal error is minimal for the knotsa
i
.) Then the adaptive methods for the same problem and methods for seeking the maximum of a Lipschitz function are considered. While adaptive methods are not better when considering the maximal error (this is valid for our examples as well as for many others) this is in general not the case with the average error. 相似文献
152.
Summary Under suitable conditions, we prove the convergence of the Bateman method for integral equations defined over bounded domains inR
d
,d1. The proof makes use of Hilbert space methods, and requires the integral operator to be non-negative definite. For one-dimensional integral equations over finite intervals, estimated rates of convergence are obtained which depend on the smoothness of the kernel, but are independent of the inhomogeneous term. In particular, for aC
kernel andn reasonably spaced Bateman points, the convergence is shown to be faster than any power of 1/n. Numerical calculations support this result. 相似文献
153.
Summary A scheme that uses singular perturbation theory to improve the performance of existing finite element methods is presented. The proposed scheme improves the error bounds of the standard Galerkin finite element scheme by a factor of O(n+1) (where is the small parameter andn is the order of the asymptotic approximation). Numerical results for linear second order O.D.E.'s are given and are compared with several other schemes. 相似文献
154.
Rüdiger Verfürth 《Numerische Mathematik》1986,50(6):697-721
Summary We consider a mixed finite element approximation of the stationary, incompressible Navier-Stokes equations with slip boundary condition, which plays an important rôle in the simulation of flows with free surfaces and incompressible viscous flows at high angles of attack and high Reynold's numbers. The central point is a saddle-point formulation of the boundary conditions which avoids the well-known Babuka paradox when approximating smooth domains by polyhedrons. We prove that for the new formulation one can use any stable mixed finite element for the Navier-Stokes equations with no-slip boundary condition provided suitable bubble functions on the boundary are added to the velocity space. We obtain optimal error estimates under minimal regularity assumptions for the solution of the continous problem. The techniques apply as well to the more general Navier boundary condition. 相似文献
155.
Robert Kosler 《Numerische Mathematik》1986,50(1):45-55
Summary Operator equationsTu=f are approximated by Galerkin's method, whereT is a monotone operator in the sense of Browder and Minty, so that existence results are available in a reflexive Banach spaceX. In a normed spaceY error estimates are established, which require a priori bounds for the discrete solutionsu
h in the norm of a suitable space
. Sufficient conditions for the uniform boundedness u
h
Z
=O(1) ash0 are proved. Well-known error estimates in [3] for the special caseX=Y=Z are generalized by this. The theory is applied to quasilinear elliptic boundary value problems of order 2m in a bounded domain
. The approximating subspaces are finite element spaces. Especially the caseX=W
0
m, p
(), 1<p<,Y=W
0
m. 2
(),Z=W
0
m. max (2,p)
()Wm, () is treated. Some examples for 1<p<2 are considered. Forp2 a refined technique is introduced in the author's paper [7]. 相似文献
156.
A. Abras A. A. G. Campos A. V. de Carvalho L. O. Ladeira 《Applied Physics A: Materials Science & Processing》1986,41(3):185-189
Iron-boride layers on low-carbon steel were produced by thermochemical diffusion process. The surface interaction products: Fe2B, FeB, FeBx (x>1) and a solid solution of iron in boron were identified by surface Mössbauer spectroscopy (CEMS and XMS). Samples of original and boronized steel were subjected to corrosion process by immersion in HCl (0.1 N) solution for 150 h. While the steel sample was strongly corroded, none corrosion product was found on the boronized sample surface. However, significant changes in relative percentages of the various iron boride phases were detected. Also, samples of original and boronized steel were subjected to oxidation process by heat-treatment in air at 300°C for 8 h and 500°C for 4 h. At 300°C, while bulk Fe3O4 and -Fe2O3 were formed on the steel surface, none iron oxide was detected on the boronized surface. At 500° C, while only pure bulk -Fe2O3 was detected on the steel surface, a particle size distribution of-Fe2O3, with particle size of about 100 Å, was probably formed on the boronized surface, as evidenced by CEMS. 相似文献
157.
N. Z. Mamadalieva Z. Saatov V. V. Kachala A. S. Shashkov 《Chemistry of Natural Compounds》2002,38(2):179-181
The new ecdysteroid 2-deoxyecdysterone-25-acetate was isolated from roots of Silene wallichiana Klotzsch. 相似文献
158.
Summary.
A coupled semilinear elliptic problem modelling an
irreversible, isothermal chemical reaction is introduced, and
discretised using the usual piecewise linear Galerkin finite element
approximation. An interesting feature of the problem is that a reaction order of
less than one gives rise to a "dead core" region. Initially,
one
reactant is assumed to be acting as a catalyst and is kept constant. It
is shown that error bounds previously obtained for a scheme involving
numerical integration can be improved upon by considering a quadratic regularisation
of the nonlinear term.
This technique is then applied to the full coupled problem, and optimal
and error bounds
are proved in the absence of
quadrature. For a scheme involving numerical integration,
bounds similar to those
obtained for the catalyst problem are shown to hold.
Received May 25, 1993 / Revised version received July 5, 1994 相似文献
159.
Summary.
For implicit RK-methods applied to
singularly perturbed systems of ODEs it is shown
that the resulting discrete systems preserve the
geometric properties of the underlying ODE. This
invariant manifold result is used to derive sharp
bounds on the global error of RK-solutions.
Received August 26, 1993 / Revised version received May 10,
1994 相似文献
160.
Summary.
In recent years, it has been shown that many modern iterative algorithms
(multigrid schemes, multilevel preconditioners, domain decomposition
methods etc.)
for solving problems resulting from the discretization
of PDEs can be
interpreted as additive (Jacobi-like) or multiplicative
(Gauss-Seidel-like) subspace correction methods. The key to their
analysis is the study of certain metric properties of the underlying
splitting of the discretization space into a sum of subspaces
and the splitting of the variational problem on into auxiliary problems on
these subspaces.
In this paper, we propose a modification of the abstract convergence
theory of the additive and multiplicative Schwarz methods, that
makes the relation to traditional iteration methods more explicit.
The analysis of the additive and multiplicative Schwarz iterations
can be carried out in almost the same spirit as in the
traditional block-matrix
situation, making convergence proofs of multilevel and domain decomposition
methods clearer, or, at least, more classical.
In addition, we present a
new bound for the convergence rate of the appropriately scaled
multiplicative Schwarz method directly in terms
of the condition number of the corresponding additive
Schwarz operator.
These results may be viewed as an appendix to the
recent surveys [X], [Ys].
Received February 1, 1994 / Revised version received August
1, 1994 相似文献