Asymptotic expansions of finite element solutions to Robin problems in H 3 and their application in extrapolation cascadic multigrid method

For the Poisson equation with Robin boundary conditions,by using a few techniques such as orthogonal expansion(M-type),separation of the main part and the finite element projection,we prove for the first time that the asymptotic error expansions of bilinear finite element have the accuracy of O(h3)for u∈H3.Based on the obtained asymptotic error expansions for linear finite elements,extrapolation cascadic multigrid method(EXCMG)can be used to solve Robin problems effectively.Furthermore,by virtue of Richardson not only the accuracy of the approximation is improved,but also a posteriori error estimation is obtained.Finally,some numerical experiments that confirm the theoretical analysis are presented.     

An asymptotic expansion of the distribution of Rao's U-statistic under a general condition

In this paper we consider the problem of testing the hypothesis about the sub-mean vector. For this propose, the asymptotic expansion of the null distribution of Rao's U-statistic under a general condition is obtained up to order of n^-1. The same problem in the k-sample case is also investigated. We find that the asymptotic distribution of generalized U-statistic in the k-sample case is identical to that of the generalized Hotelling's T² distribution up to n^-1. A simulation experiment is carried out and its results are presented. It shows that the asymptotic distributions have significant improvement when comparing with the limiting distributions both in the small sample case and the large sample case. It also demonstrates the equivalence of two testing statistics mentioned above.     

Extrapolation algorithms for solving nonlinear boundary integral equations by mechanical quadrature methods

We study the numerical solution procedure for two-dimensional Laplace’s equation subjecting to non-linear boundary conditions. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equations. Mechanical quadrature methods are presented for solving the equations, which possess high accuracy order O(h ³) and low computing complexities. Moreover, the algorithms of the mechanical quadrature methods are simple without any integration computation. Harnessing the asymptotical compact theory and Stepleman theorem, an asymptotic expansion of the errors with odd powers is shown. Based on the asymptotic expansion, the h ³ −Richardson extrapolation algorithms are used and the accuracy order is improved to O(h ⁵). The efficiency of the algorithms is illustrated by numerical examples.     

Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle

An asymptotic expansion including error bounds is given for polynomials {P _n, Q_n} that are biorthogonal on the unit circle with respect to the weight function (1?e^iθ)^α+β(1?e^?iθ)^α?β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function₂ F ₁(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.     

Corner boundary layer in nonlinear singularly perturbed elliptic problems

The Dirichlet problem in a rectangle is considered for the elliptic equation ?²Δu = F(u, x, y, ?), where F(u, x, y, ?) is a nonlinear function of u. The method of corner boundary functions is applied to the problem. Assuming that the leading term of the corner part of the asymptotics exists, an asymptotic expansion of the solution is constructed and the remainder is estimated.     

Topological and fractal properties of real numbers which are not normal

The set L of essentially non-normal numbers of the unit interval (i.e., the set of real numbers having no asymptotic frequencies of all digits in their nonterminating s-adic expansion) is studied in details. It is proven that the set L is generic in the topological sense (it is of the second Baire category) as well as in the sense of fractal geometry (L is a superfractal set, i.e., the Hausdorff-Besicovitch dimension of the set L is equal 1). These results are substantial generalizations of the previous results of the two latter authors [M. Pratsiovytyi, G. Torbin, Ukrainian Math. J. 47 (7) (1995) 971-975].The Q^∗-representation of real numbers (which is a generalization of the s-adic expansion) is also studied. This representation is determined by the stochastic matrix Q^∗. We prove the existence of such a Q^∗-representation that almost all (in the sense of Lebesgue measure) real numbers have no asymptotic frequency of all digits. In the case where the matrix Q^∗ has additional asymptotic properties, the Hausdorff-Besicovitch dimension of the set of numbers with prescribed asymptotic properties of their digits is determined (this is a generalization of the Eggleston-Besicovitch theorem). The connections between the notions of “normality of numbers” respectively of “asymptotic frequencies” of their digits is also studied.     

On a test of equality of two-parameter exponential distributions

In this paper, an asymptotic expansion of the distribution of the statistic for testing the equality of p two-parameter exponential distributions is obtained upto the order n^?4 with the second term of the order n^?3 where n is the size of the sample drawn from the i th exponential population. The asymptotic expansion can therefore be used to obtain accurate approximations to the critical values of the test statistic even for comparatively small values of n. Also we have shown that F-tables can be used to test the hypothesis when the sample size is moderately large.     

Weak Convergence of Cardaliaguet-Euvrard Neural Network Operators Studied Asymptotically

An Nth order asymptotic expansion is established for the error of weak approximation of a special class of functions by the well-known Cardaliaguet-Euvrard neural network operators. This class is made out of functions f that are N times continuously differentiable over R, so that all f,f′,…, f ^(N) have the same compact support and f ^(N) is of bounded variation. This asymptotic expansion involves products of integrals of the network activation bell-shaped function b and f. The rate of the above convergence depends only on the first derivative of involved functions.     

Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods

In this paper we analyze the stream function-vorticity-pressure method for the Stokes eigenvalue problem. Further, we obtain full order convergence rate of the eigenvalue approximations for the Stokes eigenvalue problem based on asymptotic error expansions for two nonconforming finite elements, Q ₁^rot and EQ ₁^rot. Using the technique of eigenvalue error expansion, the technique of integral identities and the extrapolation method, we can improve the accuracy of the eigenvalue approximations. This project is supported in part by the National Natural Science Foundation of China (10471103) and is subsidized by the National Basic Research Program of China under the grant 2005CB321701.     

A singularly perturbed semilinear reaction-diffusion problem in a polygonal domain

The semilinear reaction-diffusion equation −ε²Δu+b(x,u)=0 with Dirichlet boundary conditions is considered in a convex polygonal domain. The singular perturbation parameter ε is arbitrarily small, and the “reduced equation” b(x,u₀(x))=0 may have multiple solutions. An asymptotic expansion for u is constructed that involves boundary and corner layer functions. By perturbing this asymptotic expansion, we obtain certain sub- and super-solutions and thus show the existence of a solution u that is close to the constructed asymptotic expansion. The polygonal boundary forces the study of the nonlinear autonomous elliptic equation −Δz+f(z)=0 posed in an infinite sector, and then well-posedness of the corresponding linearized problem.     

Local theory of extrapolation methods

In this paper we discuss the theory of one-step extrapolation methods applied both to ordinary differential equations and to index 1 semi-explicit differential-algebraic systems. The theoretical background of this numerical technique is the asymptotic global error expansion of numerical solutions obtained from general one-step methods. It was discovered independently by Henrici, Gragg and Stetter in 1962, 1964 and 1965, respectively. This expansion is also used in most global error estimation strategies as well. However, the asymptotic expansion of the global error of one-step methods is difficult to observe in practice. Therefore we give another substantiation of extrapolation technique that is based on the usual local error expansion in a Taylor series. We show that the Richardson extrapolation can be utilized successfully to explain how extrapolation methods perform. Additionally, we prove that the Aitken-Neville algorithm works for any one-step method of an arbitrary order s, under suitable smoothness.     

Quantile Function Expansion Using Regularly Varying Functions

We present a simple result that allows us to evaluate the asymptotic order of the remainder of a partial asymptotic expansion of the quantile function h(u) as u → 0⁺ or 1^?. This is focussed on important univariate distributions when h(?) has no simple closed form, with a view to assessing asymptotic rate of decay to zero of tail dependence in the context of bivariate copulas. Motivation of this study is illustrated by the asymptotic behaviour of the tail dependence of Normal copula. The Normal, Skew-Normal and Gamma are used as initial examples. Finally, we discuss approximation to the lower quantile of the Variance-Gamma and Skew-Slash distributions.     

Two-point Taylor expansions in the asymptotic approximation of double integrals. Application to the second and fourth Appell functions

The main difficulty in Laplace's method of asymptotic expansions of double integrals is originated by a change of variables. We consider a double integral representation of the second Appell function F₂(a,b,b^′,c,c^′;x,y) and illustrate, over this example, a variant of Laplace's method which avoids that change of variables and simplifies the computations. Essentially, the method only requires a Taylor expansion of the integrand at the critical point of the phase function. We obtain in this way an asymptotic expansion of F₂(a,b,b^′,c,c^′;x,y) for large b, b^′, c and c^′. We also consider a double integral representation of the fourth Appell function F₄(a,b,c,d;x,y). We show, in this example, that this variant of Laplace's method is uniform when two or more critical points coalesce or a critical point approaches the boundary of the integration domain. We obtain in this way an asymptotic approximation of F₄(a,b,c,d;x,y) for large values of a,b,c and d. In this second example, the method requires a Taylor expansion of the integrand at two points simultaneously. For this purpose, we also investigate in this paper Taylor expansions of two-variable analytic functions with respect to two points, giving Cauchy-type formulas for the coefficients of the expansion and details about the regions of convergence.     

A second-order estimate for blow-up solutions of elliptic equations

We investigate second-term asymptotic behavior of boundary blow-up solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, in a bounded smooth domain Ω⊂R^N. b(x) is a non-negative weight function. The nonlinearly f is regularly varying at infinity with index ρ>1 (that is lim_u→∞f(ξu)/f(u)=ξ^ρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). The main results show how the mean curvature of the boundary ∂Ω appears in the asymptotic expansion of the solution u(x). Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.     

Asymptotic expansions for classical and generalized divided differences including applications

It is a well-known fact that the classical (i.e. polynomial) divided difference of orderm, when applied to a functiong, converges to themth-derivative of this function, if the evaluation points all collapse to a single one.In the first part of this paper we shall sharpen this result in the sense that we prove the existence of an asymptotic expansion with limitg ^(m) /m!. This result allows the application of extrapolation methods for the numerical differentiation of funtions.Moreover, in the second and main part of the paper we study generalized divided differences, which were introduced by Popoviciu [10] and further investigated for example by Karlin [2], Walz [15] and, mainly, Mühlbach [6–8]; we prove the existence of an asymptotic expansion also for these generalized divided differences, if the underlying function space is a Polya space. As a by-product, our results show that the generalized divided difference of orderm converges to the value of a certainmth order differential operator.     

Asymptotic expansions in time for solutions of Schrödinger-type equations

Let ?iA = ?i(p(D) + V) be a dissipative operator in L₂(R_n), where p(D) is an elliptic differential operator of order m with real constant coefficients and V is a compact operator from the weighted Sobolev space H^m′,s (Rⁿ) to H^{0, p + s} (Rⁿ), s?R, for some m′ < m ? 1 and p > 1. Let R(z) be the resolvent of A. Then an asymptotic expansion of R(z) as z approaches a critical value of the polynomial p(ξ) is given; the coefficient operators in the expansion are computed explicitly. By using the resolvent expansion and the results of M. Murata [9], an asymptotic expansion of e^?itA as t → ∞ is given.     

On measurement of electrical conductivity

A complete uniform asymptotic expansion is found for the integral ?_S ?² udxdy, where u is the solution of the Neumann problem with a delta-function-like derivative on the boundary. A physical application of the result is discussed.     

Asymptotic expansions for the variance of stopping times in nonlinear renewal theory

We treat the problem of finding asymptotic expansions for the variance of stopping times for Wiener processes with positive drift (continuous time case) as well as sums of i.i.d. random variables with positive mean (discrete time case). Carrying over the setting of nonlinear renewal theory to Wiener processes, we obtain an asymptotic expansion up to vanishing terms in the continuous time case. Applying the same methods to sums of i.i.d. random variables, we also provide an expansion in the discrete time case up to terms of order o(b^1/2) where the leading term is of order O(b), as b → ∞. The possibly unbounded term is the covariance of nonlinear excess and stopping time.     

Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots

An asymptotic expansion for large sample size n is derived by a partial differential equation method, up to and including the term of order n^?2, for the ₀F₀ function with two argument matrices which arise in the joint density function of the latent roots of the covariance matrix, when some of the population latent roots are multiple. Then we derive asymptotic expansions for the joint and marginal distributions of the sample roots in the case of one multiple root.     

Weighted stationary phase of higher orders

The subject matter of this paper is an integral with exponential oscillation of phase f(x) weighted by g(x) on a finite interval [α β]: When the phase f(x) has a single stationary point in (α β), an nth-order asymptotic expansion of this integral is proved for n ≥ 2: This asymptotic expansion sharpens the classical result for n = 1 by M. N. Huxley. A similar asymptotic expansion was proved by V. Blomer, R. Khan and M. Young under the assumptions that f(x) and g(x) are smooth and g(x) is compactly supported on R: In the present paper, however, these functions are only assumed to be continuously differentiable on [α β] 2n + 3 and 2n + 1 times, respectively. Because there are no requirements on the vanishing of g(x) and its derivatives at the endpoints α and β, the present asymptotic expansion contains explicit boundary terms in the main and error terms. The asymptotic expansion in this paper is thus applicable to a wider class of problems in analysis, analytic number theory, and other fields.