Finite-Element Approximation of Elliptic Equations with a Neumann or Robin Condition on a Curved Boundary

This paper considers a finite-element approximation of a second-orderself adjoint elliptic equation in a region Rⁿ (with n=2 or 3)having a curved boundary on which a Neumann or Robin conditionis prescribed. If the finite-element space defined over , a union of elements, has approximation power h^kin the L² norm, and if the region of integration is approximatedby ^h with dist (, ^h)Ch^k, then it is shown that one retains optimalrates of convergence for the error in the H¹ and L² norms, whetherQ^h is fitted or unfitted , provided that the numerical integration scheme has sufficientaccuracy.     

Generalized compound quadrature formulae for finite-part integrals

Received on 31 July 1995. Revised on 19 August 1996. We investigate the error term of the dth degree compound quadratureformulae for finite-part integrals of the form where and p 1.We are mainly interested in error bounds of the form with best possible constants c. Itis shown that, for and n uniformlydistributed nodes, the error behaves as O(n^p–s–1for , p–1 <s d+1.In a previous paper we have shown that this is not true for As an improvement, we consider the case of non-uniformly distributednodes. Here, we show that for all p I and , an O(n^–s) error estimate can be obtainedin theory by a suitable choice of the nodes. A set of nodeswith this property is staled explicitly. In practice, this gradedmesh causes stability problems which are computationally expensiveto overcome. E-mail address: diethelm{at}informatik.uni-hildesheim.de     

Finite-Element Approximation of a Plasma Equilibrium Problem

The plasma problem studied is: given R⁺ find (, d, u) R ?R ? H¹() such that Let ₁ < ₂ be the first two eigenvalues of the associatedlinear eigenvalue problem: find $$\left(\lambda ,\phi \right)\in\mathrm{R;}\times {\hbox{ H }}_{0}^{1}\left(\Omega \right)$$such that For ₀(0,₂) it is well known that there exists a unique solution(₀, d₀, u₀) to the above problem. We show that the standard continuous piecewise linear Galerkinfinite-element approximatinon $$\left({\lambda }_{0},{\hbox{d }}_{0}^{k},{u}_{0}^{h}\right)$$, for ₀(0,₂), converges atthe optimal rate in the H¹, L², and L norms as h, the mesh length,tends to 0. In addition, we show that dist (, ^h)Ch² ln 1/h,where $${\Gamma }^{\left(h\right)}=\left\{x\in \Omega :{u}_{0}^{\left(h\right)}\left(x\right)=0\right\}$$.Finally we consider a more practical approximation involvingnumerical integration.     

An Efficient Enthalpy-type Method for the Stefan Problem

The paper presents a new finite-difference method for solvingthe one-dimensional two-phase Stefan problem. Under assumptionson the data which guarantee the temperature u and the movingboundary s to belong to and , respectively, we obtain L₂-errorestimates of order O(h + h^–?) provided the time step is chosen such that Numerical aspects are discussed.     

Stability Analysis of Runge-Kutta Methods for Volterra Integral Equations of the Second Kind

Stability analysis of Volterra-Runge-Kutta methods based onthe basic test equation of the form where is a complex parameter, and on the convolution test equation where and are real parameters, is presented. General stabilityconditions are derived and applied to construct numerical methodswith good stability properties. In particular, a family of second-orderV^o-stable Volterra-Runge-Kutta methods is obtained. No V^o-stablemethods of order greater than one have been presented previouslyin the literature.     

On Mason's conjecture concerning interpolation by polynomials in z and z-1 on an annulus

A polynomial of degree n in z^–1 and n – 1 in z isdefined by an interpolation projection from the space A(N) of functions f analytic in thecircular annulus ^–1<|z| and continuous on its boundaries|z|=^–1, . The points of interpolation are chosen to coincidewith the n roots of zⁿ=–n and the n roots of zⁿ=^–n.We prove Mason's conjecture that the corresponding Lebesguefunction attains its maximal value on the inner circle. We alsoestimate the bound of the Lebesgue constant . It is proved that the following estimate for theoperator norm holds: where _n, is the Lebesgue constant of Gronwall for equally spacedinterpolation on a circle by a polynomial of degree n.     

Maximum Precision in Elliott-Donaldson Formulae

This paper considers the approximation of by quadratures which are constructed inthe manner described by Elliott & Donaldson. Results areestablished for choosing the nodes so that the quadrature is exact on as wide a range of positive and negativepowers of z as possible.     

An Order of Convergence for Some Radial Basis Functions

The quasi-interpolant to a function f : RⁿR on an infinite regulargrid of spacing h can be defined by where : RⁿR is a function which decays quickly for large argument.In the case of radial basis functions has the form where : R⁺R is known as a radial basis function and, in general,?_j R (j = 1,...,m) and x^j Rⁿ (j = 1,...,m), though here onlythe particular case x^j Zⁿ (j = 1,..., m) is considered. Thispaper concentrates on the case (r) = r, a generalization oflinear interpolation, although some of the analysis is moregeneral. It is proved that, if n is odd, then there is a function such that the maximum difference between a sufficiently smoothfunction and its quasi-interpolant is bounded by a constantmultiple of hⁿ⁺¹. This is done by first showing that such aquasi-interpolation formula can reproduce polynomials of degreen.     

Local error estimates for radial basis function interpolation of scattered data

Introducing a suitable variational formulation for the localerror of scattered data interpolation by radial basis functions(r), the error can be bounded by a term depending on the Fouriertransform of the interpolated function f and a certain ‘Krigingfunction’, which allows a formulation as an integral involvingthe Fourier transform of . The explicit construction of locallywell-behaving admissible coefficient vectors makes the Krigingfunction bounded by some power of the local density h of datapoints. This leads to error estimates for interpolation of functionsf whose Fourier transform f is ‘dominated’ by thenonnegative Fourier transform of (x) = (||x||) in the sense . Approximation orders are arbitrarily high for interpolationwith Hardy multiquadrics, inverse multiquadrics and Gaussiankernels. This was also proven in recent papers by Madych andNelson, using a reproducing kernel Hilbert space approach andrequiring the same hypothesis as above on f, which limits thepractical applicability of the results. This work uses a differentand simpler analytic technique and allows to handle the casesof interpolation with (r) = r^s for s R, s > 1, s 2N, and(r) = r^s log r for s 2N, which are shown to have accuracy O(h^s/2)     

Ergodicity of a Class of Cocycles Over Irrational Rotations

It is proved that if is irrational and L²(S¹) with o(l/n)then for each mZ\{0} the corresponding skew product is ergodic. The rigidity of specialflows over irrational rotations with roof functions whose Fouriercoefficients are in o(l/n) is also shown.     

Global Branch of Solutions for Non-Linear Schrodinger Equations with Deepening Potential Well

We consider the stationary non-linear Schrödinger equation where > 0 and the functionsf and g are such that and for some bounded open set R^N. We use topological methods to establish the existenceof two connected sets D^± of positive/negative solutionsin R x W², ^p R^N where that cover the interval (, ()) in the sense that and furthermore, The number () is characterized as the unique value of in theinterval (, ) for which the asymptotic linearization has a positiveeigenfunction. Our work uses a degree for Fredholm maps of indexzero. 2000 Mathematics Subject Classification 35J60, 35B32,58J55.     

Finite Element Approximation of a Rigid Punch Indenting a Membrane

Optimal order H¹ and L error bounds are obtained for a continuouspiecewise linear finite element approximation of an obstacleproblem, where the obstacle's height as well as the contactzone, _c, are a priori unknown. The problem models the indentationof a membrane by a rigid punch. For R², given ,g R⁺ and an obstacle defined over E we consider the minimization of |v|²_1,+gµover (v, µ) H¹₀() x R subject to v+µ on E. In additionwe show under certain nondegeneracy conditions that dist (_c,^h_c)Ch ln 1/h, where ^h_c is the finite element approximation to_c. Finally we show that the resulting algebraic problem canbe solved using a projected SOR algorithm.     

Some Relations Between Packing Premeasure and Packing Measure

Let K be a compact subset of Rⁿ, 0 s n. Let , P^s denote s-dimensional packing premeasure andmeasure, respectively. We discuss in this paper the relationbetween and P^s. We prove:if , then ; and if , then for any > 0, there exists a compact subset F of K such that and P^s(F) P^s(K) – .1991 Mathematics Subject Classification 28A80, 28A78.     

Convergence of Univariate Quasi-Interpolation Using Multiquadrics

Quasi-interpolants to a function f: RR on an infinite regularmesh of spacing h can be defined by where :RR is a function with fast decay for large argument. In the approach employing the radial-basis-function : RR, thefunction is a finite linear combination of basis functions(|•–jh|) (jZ). Choosing Hardy's multiquadrics (r)=(r²+c²)^?,we show that sufficiently fast-decaying exist that render quasi-interpolationexact for linear polynomials f. Then, approximating f C²(R),we obtain uniform convergence of s to f as (h, c)0, and convergenceof s' to f' as (h, c²/h)0. However, when c stays bounded awayfrom 0 as h0, there are f C(R) for which s does not convergeto f as h0. We also show that, for all which vanish at infinity but arenot integrable over R, there are no finite linear combinations of the given basis functions allowing the construction of admissiblequasi-interpolants. This includes the case of the inverse multiquadncs(r)=(r²+c²)^–?.     

Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems I: the scalar case

^** Email: Paul.Houston{at}mcs.le.ac.uk^*** Email: Janice.Robson{at}comlab.ox.ac.uk^**** Email: Endre.Suli{at}comlab.ox.ac.uk We develop a one-parameter family of hp-version discontinuousGalerkin finite element methods, parameterised by [–1,1], for the numerical solution of quasilinear elliptic equationsin divergence form on a bounded open set ^d, d 2. In particular,we consider the analysis of the family for the equation –·{µ(x, |u|)u} = f(x) subject to mixed Dirichlet–Neumannboundary conditions on . It is assumed that µ is a real-valuedfunction, µ C( x [0, )), and thereexist positive constants m_µ and M_µ such that m_µ(t– s) µ(x, t)t – µ(x, s)s M_µ(t– s) for t s 0 and all x . Using a result from the theory of monotone operators for any valueof [–1, 1], the corresponding method is shown to havea unique solution u_DG in the finite element space. If u C¹() H^k(), k 2, then with discontinuous piecewise polynomials ofdegree p 1, the error between u and u_DG, measured in the brokenH¹()-norm, is (h^s–1/p^k–3/2), where 1 s min {p+ 1, k}.     

Determinants of Lattices induced by Rational Subspaces

Let L and be orthogonal complementary rational linear subspaces of Eⁿ, and let = L Zⁿ and $$\stackrel{\&macr;}{\Lambda}$$ = Zⁿ be the sublatticesof the usual integer lattice Zⁿ induced by L and . Then the determinants of and are equal. The samerelationship holds between the determinants of the lattices and obtained by orthogonal projection of Zⁿ on to L and .     

The Effects of Rounding Error on an Algorithm for Downdating a Cholesky Factorization

Let the positive definite matrix A have a Cholesky factorizationA= R^TR. For a given vector xsuppose that ? =A - xx^T has a Choleskyfactorization ? = ^T.This paper considers an algorithm for computing from R and x and an extension for removing a row from the QR factorizationof a regression problem. It is shown that the algorithm is stablein the presence of rounding errors. However, it is also shownthat the matrix can be a very ill-conditioned function of R and x.     

Reducing Subspaces for a Class of Multiplication Operators

Let D be the open unit disk in the complex plane C. The Bergmanspace is the Hilbert space of analytic functions f in D such that where dA is the normalized area measure on D. If are two functions in , then the inner product of f and g is given by We study multiplication operators on induced by analytic functions. Thus for H (D), the space ofbounded analytic functions in D, we define by It is easy to check that M is a bounded linear operator on with ||M||=||||=sup{|(z)|:zD}.     

Regular approximation of singular self-adjoint differential operators

Given a singular self-adjoint differential operator of order 2n with real coefficients we constructtwo sequences of regular self-adjoint differential expressions^r which converge to ina generalized sense of resolvent convergence. The first constructionis suitable when no information about the real resolvent setof is available. The second is suitablewhen we know a real point of the resolvent set of .The main application of this construction is in numerical solutionof singular differential equations.     

Oscillation Criteria for First-Order Delay Equations

This paper is concerned with the oscillatory behaviour of first-orderdelay differential equations of the form (1) where is non-decreasing, (t)< t for t t₀ and . Let the numbers k andL be defined by It is proved here that when L < 1 and 0 < k 1/e all solutionsof equation (1) oscillate in several cases in which the condition holds, where ₁ is the smaller root of the equation = e^k. 2000Mathematics Subject Classification 34K11 (primary); 34C10 (secondary).