Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data

Summary In this paper we study the convergence properties of a fully discrete Galerkin approximation with a backwark Euler time discretization scheme. An approach based on semigroup theory is used to deal with the nonsmooth Dirichlet boundary data which cannot be handled by standard techniques. This approach gives rise to optimal rates of convergence inL _p[O,T;L ₂()] norms for boundary conditions inL _p[O,T;L ₂()], 1p.     

Numerical application of Euler's series transformation and its generalizations

Summary A nonlinear generalizationÊ _z of Euler's series transformation is compared with the (linear) Euler-Knopp transformationE _z and a twoparametric methodE . It is shown how to applyE orE _, to compute the valuef(z_o) of a functionf from the power series at 0 iff is holomorphic in a half plane or in the cut plane. BothE andE _, are superior toÊ _z . A compact recursive algorithm is given for computingE andE _,.     

Nonlinear stability and convergence of finite-difference methods for the “good” Boussinesq equation

Summary The good Boussinesq equationu _tt =–u _xxxx +u _xx +(u ²) _xx has recently been found to possess an interesting soliton-interaction mechanism. In this paper we study the nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the good Boussinesq equation. Numerical experimentas are also reported.     

The dependence of critical parameter bounds on the monotonicity of a Newton sequence

Summary A new method is proposed for the inclusion of the critical parameter ^* of some convex operator equationu=Tu (appearing e.g. in thermal explosion theory). It is based on the fact that for a fixed Newton's method starting with a suitable subsolution is not monotonically if and only if >^*. Several numerical examples arising from nonlinear boundary value problems illustrate the efficiency of the method.     

Stability of a spline collocation method for strongly elliptic multidimensional singular integral equations

Summary We consider a spline collocation method for strongly elliptic zero order pseudodifferential equationsp _gw Au=f on a cube =(0, 1) ^m . Utilizing multilinear spline functions which are zero at the boundary we collocate at the meshpoints inside . For classical strongly elliptic translation invariant pseudodifferential operators, we verify the stability of the considered collocation method inL ²(). Afterwards, form2 and a right hand sidefH ⁸(),s>m/2, we prove an asymptotic convergence estimate.The author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant number Ko 634/32-1     

Rates of convergence and optimal spectral bandwidth for long range dependence

Summary For a realization of lengthn from a covariance stationary discrete time process with spectral density which behaves like ^1–2H as 0+ for 1/2<H<1 (apart from a slowly varying factor which may be of unknown form), we consider a discrete average of the periodogram across the frequencies 2j/n,j=1,..., m, wherem andm/n0 asn. We study the rate of convergence of an analogue of the mean squared error of smooth spectral density estimates, and deduce an optimal choice ofm.     

A spectral Galerkin approximation of the Orr-Sommerfeld eigenvalue problem in a semi-infinite domain

Summary The convergence of a Galerkin approximation of the Orr-Sommerfeld eigenvalue problem, which is defined in a semi-infinite domain, is studied theoretically. In case the system of trial functions is based on a composite of Jacobi polynomials and an exponential transform of the semi-infinite domain, the error of the Galerkin approximation is estimated in terms of the transformation parametera and the numberN of trial functions. Finite or infinite-order convergence of the spectral Galerkin method is obtained depending on how the transformation parameter is chosen. If the transformation parameter is fixed, then convergence is of finite order only. However, ifa is varied proportional to 1/N with an exponent 0<<1, then the approximate eigenvalue converges faster than any finite power of 1/N asN. Some numerical examles are given.     

A θ-stable approximations of abstract Cauchy problems

Summary We study the approximation of linear parabolic Cauchy problems by means of Galerkin methods in space andA -stable multistep schemes of arbitrary order in time. The error is evaluated in the norm ofL _t ² (H _x ¹ ) L _t (L _x ² ).     

An asymptotic finite element method for improvement of solutions of boundary layer problems

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.     

Quadratic spline interpolation on uniform meshes

The interpolation problem at uniform mesh points of a quadratic splines(x _i)=f _i,i=0, 1,...,N ands(x ₀)=f₀ is considered. It is known that s–f=O(h ³) and s–f=O(h ²), whereh is the step size, and that these orders cannot be improved. Contrary to recently published results we prove that superconvergence cannot occur for any particular point independent off other than mesh points wheres=f by assumption. Best error bounds for some compound formulae approximatingf _i andf _i ⁽³⁾ are also derived.     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

On a numerical approach to Stefan-like problems

Summary This paper is devoted to the numerical analysis of a bidimensional two-phase Stefan problem. We approximate the enthalpy formulation byC ⁰ piecewise linear finite elements in space combined with a semi-implicit scheme in time. Under some restrictions related to the finite element mesh and to the timestep, we prove positivity, stability and convergence results. Various numerial tests are presented and discussed in order to show the accuracy of our scheme.This work is supported by the Fonds National Suisse pour la Recherche Scientifique.     

On the relation between Fourier and Leont’ev coefficients with respect to smirnov spaces

Yu. Melnik showed that the Leontev coefficients _f () in the Dirichlet series 2}}$$ " align="middle" border="0"> of a function f E ^p (D), 1 < p < , are the Fourier coefficients of some function F L ^p , ([0, 2]) and that the first modulus of continuity of F can be estimated by the first moduli and majorants in f. In the present paper, we extend his results to moduli of arbitrary order.Published in Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 517–526, April, 2004.     

Reflected Brownian motion in a cone: Semimartingale property

Summary In this paper, the object of study is reflected Brownian motion in a cone ind-dimensions (d3) with nonconstant oblique reflection on each radial line emanating from the vertex of the cone. The basic question considered here is When is this process a semimartingale?. Conditions for the existence and uniqueness of the process for which the vertex is an instantaneous state were given by Kwon, which is resolved in terms of a real parameter depending on the cone and the direction of reflection. It is shown that starting from any point of the cone, the process is a semimartingale if < 1, + 0 and not a semimartingale if < < 2.This research is supported by KOSEF grant 941-0100-011-1     

Stability radius of polynomials occurring in the numerical solution of initial value problems

This paper deals with polynomial approximations(x) to the exponential function exp(x) related to numerical procedures for solving initial value problems. Motivated by stability requirements, we present a numerical study of the largest diskD()={z C: |z+|} that is contained in the stability regionS()={z C: |(z)|1}. The radius of this largest disk is denoted byr(), the stability radius. On the basis of our numerical study, several conjectures are made concerningr _m,p=sup {r(): _m,p}. Here _{m, p} (1pm; p, m integers) is the class of all polynomials(x) with real coefficients and degree m for which(x)=exp(x)+O(x ^p+1) (forx 0).     

A relation between the zeros of an <Emphasis Type="Italic">L</Emphasis>-function belonging to the Selberg class and the zeros of an associated <Emphasis Type="Italic">L</Emphasis>-function twisted by a Dirichlet character

Let F (s) be a function belonging to the Selberg class. For a primitive Dirichlet character , we can define the -twist F(s) of F (s). If F(s) also belongs to the Selberg class and satisfies some other conditions then there is a relation between the zeros of F (s) and the zeros F(s). Further we give an operator theoretic interpretation of this relation according to A. Connes study.Received: 5 January 2004     

A regularity result for critical points of conformally invariant functionals

Let be an open set inR ² andI be a conformally invariant functional defined onH ¹(,R ^d ). Letu be a critical point ofI. We show that, ifu is apriori assumed to be bounded, thenu is smooth in , up to (ifu _| is smooth). This is a partial (positive) answer to a conjecture of S. Hildebrandt [13]. As an application, we establish a regularity result for weak solutions to the equation of surfaces of prescribed mean curvature in a three-dimensional compact riemannian manifold.     

On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323     

Interior penalty method for the indefinite time-harmonic Maxwell equations

Summary In this paper, we introduce and analyze the interior penalty discontinuous Galerkin method for the numerical discretization of the indefinite time-harmonic Maxwell equations in the high-frequency regime. Based on suitable duality arguments, we derive a-priori error bounds in the energy norm and the L²-norm. In particular, the error in the energy norm is shown to converge with the optimal order (h^min{s,}) with respect to the mesh size h, the polynomial degree , and the regularity exponent s of the analytical solution. Under additional regularity assumptions, the L²-error is shown to converge with the optimal order (h⁺¹). The theoretical results are confirmed in a series of numerical experiments.Supported by the EPSRC (Grant GR/R76615).Supported by the Swiss National Science Foundation under project 21-068126.02.Supported in part by the Natural Sciences and Engineering Council of Canada.     

Corrigendum: “On the Exceptional Set for Goldbach’s Problem in Short Intervals”

We correct the proof of the log-free hybrid zero-density estimate given in On the Exceptional Set for Goldbachs Problem in Short Intervals, Monatsh Math 132: 49–65.