Total Flux Estimates for a Finite-Element Approximation of Elliptic Equations

An elliptic boundary-value problem on a domain with prescribedDirichlet data on _I is approximated using a finite-elementspace of approximation power h^K in the L² norm. It is shownthat the total flux across _I can be approximated with an errorof O(h^K) when is a curved domain in Rⁿ (n = 2 or 3) and isoparametricelements are used. When is a polyhedron, an O(h^2K–2)approximation is given. We use these results to study the finite-elementapproximation of elliptic equations when the prescribed boundarydata on _I is the total flux. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton, Sussex BN1 9QH.     

Fitted and Unfitted Finite-Element Methods for Elliptic Equations with Smooth Interfaces

This paper considers the finite-element approximation of theelliptic interface problem: -?(u) + cu = f in Rⁿ (n = 2 or3), with u = 0 on , where is discontinuous across a smoothsurface in the interior of . First we show that, if the meshis isoparametrically fitted to using simplicial elements ofdegree k - 1, with k 2, then the standard Galerkin method achievesthe optimal rate of convergence in the H¹ and L² norms overthe approximations _l⁴ of _l where _{l 2}. Second, since itmay be computationally inconvenient to fit the mesh to , weanalyse a fully practical piecewise linear approximation ofa related penalized problem, as introduced by Babuska (1970),based on a mesh that is independent of . We show that, by choosingthe penalty parameter appropriately, this approximation convergesto u at the optimal rate in the H¹ norm over _l⁴ and in the L²norm over any interior domain _l^* satisfying _l^* _l^** _l⁴ for somedomain _l^**. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton BN1 9QH     

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.     

Realistic a Priori and a Posteriori Error Bounds for Computed Eigenvalues

Permanent address: Department of Engineering Mathematics, Cairo University, Giza, Egypt. A priori and a posteriori error bounds are given for the computedeigenpair (, ) of the eigenvalue problem Ax = x, which are shownto be more realistic than some of the available ones. A simplemethod is also presented for computing the backward error. Finallya scaling procedure is explained for reducing the residual error.     

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}.     

A new non-conforming Petrov-Galerkin finite-element method with triangular elements for a singularly perturbed advection-diffusion problem

A new non-conforming exponentially fitted Petrov-Galerkin finite-elementmethod based on Delaunay triangulation and its Dirichlet tessellationis constructed for the numerical solution of singularly perturbedstationary advectiondiffusion problems with a singular perturbationparameter . The method is analyzed mathematically and its stabilityis shown to be independent of . The error estimate in an -independentdiscrete energy norm for the approximate solution is shown todepend on first-order seminorms of the flux and the zero-orderterm of the equation, the sup norm of the exact solution, thefirst-order seminorm of the coefficient of the advection term,and the approximation error of the inhomogeneous term. Sincethe first two seminorms are not bounded uniformly in , the -uniformconvergence of the method still remains an open question. Noassumption is required that the angles in the triangulationare all acute. Since the system matrix for this method is identicalto that for the exponentially fitted box method, the theoreticalresults also provide an analysis of that box method. The newmethod also contains the central-difference and upwind methodsas two limiting cases. It can be regarded as a weighted finite-differencemethod on a triangular mesh. Numerical results are presentedto show the superior performance of the method in comparisonwith the upwind and central-difference methods for a small increasein the computation cost. Present address: School of Mathematics, The University of NewSouth Wales, Kensington, NSW 2033, Australia.     

Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces

We consider a fully practical finite-element approximationof the following system of nonlinear degenerate parabolic equations: (u)/(t) + . (u² [(v)]) - (1)/(3) .(u³ w)= 0, w = - c u - u^-+ a u^-3 , (v)/(t) + . (u v [(v)]) - v - .(u² v w) = 0. The above models a surfactant-driven thin-film flow in the presenceof both attractive, a>0, and repulsive, >0 with >3,van der Waals forces; where u is the height of the film, v isthe concentration of the insoluble surfactant monolayer and(v):=1-v is the typical surface tension. Here 0 and c>0 arethe inverses of the surface Peclet number and the modified capillarynumber. In addition to showing stability bounds for our approximation,we prove convergence, and hence existence of a solution to thisnonlinear degenerate parabolic system, (i) in one space dimensionwhen >0; and, moreover, (ii) in two space dimensions if inaddition 7. Furthermore, iterative schemes for solving the resultingnonlinear discrete system are discussed. Finally, some numericalexperiments are presented.     

Boundary integral methods for singularly perturbed boundary value problems

In this paper we consider boundary integral methods appliedto boundary value problems for the positive definite Helmholtz-typeproblem –U + ²U = 0 in a bounded or unbounded domain,with the parameter real and possibly large. Applications arisein the implementation of space–time boundary integralmethods for the heat equation, where is proportional to 1/(t),and t is the time step. The corresponding layer potentials arisingfrom this problem depend nonlinearly on the parameter and havekernels which become highly peaked as , causing standard discretizationschemes to fail. We propose a new collocation method with arobust convergence rate as . Numerical experiments on a modelproblem verify the theoretical results.     

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)^–?.     

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.     

Total Flux Estimates for a Finite-Element Approximation of Parabolic Equations

An initial-boundary-value problem for a parabolic equation ina domain x (0, T) with prescribed Dirichlet data on is approximatedusing a continuous-time Galerkin finite-element scheme. It isshown that the total flux across ₁= can be approximated withan error of O(h^k) when is a curved domain in Rⁿ (n = 2 or 3)and isoparametric elements having approximation power h^k inthe L² norm are used.     

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     

The Convergence of Operator Approximations at Turning Points

Present address: Department of Mathematics, University of Reading, Reading RG6 2AX. We consider the convergence of solution curves of approximationsto parameter-dependent operator equations of the form G(, x)= 0. Provided G_x(, x) remains non-singular this problem is cateredfor by a simple extension to standard theory. In this paper,however, attention is concentrated on solution curves throughcertain singular points (⁰, x⁰), and the main result is thatconvergence depends on consistency and stability results forthe linear eigenvalue problem G_x(⁰, x⁰)0 = 0.     

Delay dependent stability regions of {Theta}-methods for delay differential equations

In this paper asymptotic stability properties of -methods fordelay differential equations (DDEs) are considered with respectto the test equation y'(t) = ay(t) + by(t - ), t > 0, y(t)= g(t), - t 0, where > 0. First we examine extensivelythe instance where a, b and g(t) is a continuous real-valuedfunction; then we investigate the more general case of a, b C and g(t) a continuous complex-valued function. The last decade has seen a relatively large number of papersdevoted to the study of the stability of -methods, using thetest equation (0.1). In those papers, conditions that are strongerthan necessary for the (asymptotic) stability of the zero solutionare assumed; for instance, [a]+¦b¦ < 0, thatis the set of complex pairs (a, b) such that the zero solutionof (0.1) is asymptotically stable for every > 0. In thispaper we study, instead, the stability properties of -methodsfor equation (0.1) with an arbitrary but fixed value of .     

Error Bounds for Hermite Interpolation by Quadratic Splines on an {alpha}-Triangulation

For l, an -triangulation F of a planar domain is such that,for every T F, there holds 1 R_T/2r_T , where R_T (resp. r_T)denotes the radius of the circumscribed (resp. inscribed) circleof the triangle T. When T is varying in F the centre of itsinscribed circle is varying in a compact interior to T and itsorthogonal projections on the sides are varying in compact intervalsinterior to these sides. Precise results are given about thesizes of these compacts and are used for the computation oferror constants in the problem of Hermite interpolation by Powell-Sabinquadratic finite elements, bringing to the fore their dependenceon the parameter .     

Multiplicative decomposability of shifted sets

The following two problems are open.

1. Do two sets of positiveintegers and exist, with at leasttwo elements each, suchthat + coincides with the set of primes for sufficiently largeelements?
2. Let ={6, 12, 18}. Is there an infinite set of positiveintegerssuch that +1? A positive answer would imply that thereare infinitelymany Carmichael numbers with three prime factors.

In this paper we prove the multiplicative analogue of the firstproblem, namely that there are no two sets and , with at leasttwo elements each, such that the product coincides with anyadditively shifted copy +c of the set of primes for sufficientlylarge elements. We also prove that shifted copies of sets ofintegers that are generated by certain subsets of the primescannot be multiplicatively decomposed.     

On spurious asymptotic numerical solutions of explicit Runge-Kutta methods

The bifurcation diagram associated with the logistic equationⁿ⁺¹ = aⁿ(1 – ⁿ) is by now well known, as is its equivalenceto solving the ordinary differential equation (ODE) u' = u(1– u) by the explicit Euler difference scheme. It has alsobeen noted by Iserles that other popular difference schemesmay not only exhibit period doubling and chaotic phenomena butalso possess spurious fixed points. We investigate, both analyticallyand computationally, Runge-Kutta schemes applied to the equationu'=f(u), for f(u) = u{1 – u) and f(u) = au(1 – u)(b– u), contrasting their behaviour with the explicit Eulerscheme. We determine and provide a local analysis of bifurcationsto spurious fixed points and periodic orbits. In particularwe show that these may appear below the linearised stabilitylimit of the scheme, and may consequently lead to erroneouscomputational results. Major part of the material was published as an internal report-NASATechnical Memorandum 102919, April 1990, also as Universityof Reading Numerical Analysis Report 3/90, March 1990. This work was performed whilst a visiting scientist at NASAAmes Research Center, Moffett Field. CA 94035 USA. Staff Scientist, Fluid Dynamics Division.     

Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind

Much simplified expressions for certain complete elliptic integralsin terms of the beta function are produced. The reverse procedureof expressing gamma functions in terms of complete ellipticintegrals allows us to use the arithmetic-geometric mean iterationto compute gamma functions, using quadratically convergent iterations.Explicit algorithms are given for computing (n/4) and (n/6),where 1m3 and 1n5.     

Discrete methods in the study of an inverse problem for Laplace's equation

Let u be harmonic in the interior of a rectangle and satisfythe third-kind boundary condition u_n + yu = where 0, y 0with supports included in the bottom and in the top side of, respectively. Recovering y from a knowledge of and of thetrace of u on the bottom is a nonlinear inverse problem ofinterest in the field of nondestructive evaluation. A convergentGalerkin method for approximating y is proposed and tested innumerical experiments.     

SMALL C*-ALGEBRAS THAT FAIL TO HAVE SEPARABLE REPRESENTATIONS

It is shown that any -finite, wild AW*-algebra of type II₁ failsto have non-trivial separable representations. Here a wild AW*-algebrameans an AW*-algebra with no direct summand, which is *-isomorphicto a von Neumann algebra. Let be the -finite, wild type II₁AW*-algebra formed by taking the monotone complete tensor productof the Dixmier algebra and a factor of type II₁ acting ona separable Hilbert space. As both and act on separable Hilbertspaces, it seems natural to describe as small and to ask thefollowing question. Does have a non-trivial separable representation?The above described result answers negatively to this question.That is, fails to have non-trivial separable representations.