Set convergence for discretizations of the attractor

We consider the discretization of a dynamical system given bya C⁰-semigroup S(t), defined on a Banach space X, possessingan attractor . Under certain weak assumptions, Hale, Lin andRaugel showed that discretizations of S(t) possess local attractors,which may be considered as approximations to . Without furtherassumptions, we show that these local attractors possess convergentsubsequences in the Hausdorff or set metric, whose limit isa compact invariant subset of . Using a new construction, wealso consider the Kloeden and Lorenz concept of attracting setsin a Banach space, and show under mild assumptions that discretizationspossess attracting sets converging to in the Hausdorff metric. ath{at}maths.bath.ac.uk Endre.Suli{at}comlab.ox.ac.uk     

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     

Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems

The long-time behaviour of continuous time Galerkin (CTG) approximationsof some well-known one-dimensional non-linear evolution problemswhich model phase transitions are analyzed. These numericalschemes are fully discrete and of arbitrary order. Partially supported by the Army Research Office through grant28535-MA. Partially supported by the ONR Contract No N00014-90-J-1238.     

On the Convergence of Some Approximate Methods of Conformal Mapping

A method was given in Ellacott (1978) for determining approximatelythe conformal mapping of a Jordan region on to a disc. Someresults on the convergence of this method are given, which canbe used to prove the result (conjectured in Ellacott, 1978)that if the boundary curve is analytic, then convergence isuniform. The corresponding result is also proved for the Bergmanand Szeg Kernel methods with polynomial basis functions. (Theresult is already known for the Szeg? Kernel method, but a differentproof is given.) Also discussed is the use of rational basisfunctions for the Bergman Kernel method. Currently visiting Forschungsinstitut fr Mathematik, ETH-Zentrum,CH-8092, Zurich.     

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.     

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.     

Nystrm interpolants based on the zeros of Legendre polynomials for a non-compact integral operator equation

We consider two different Nystrm interpolants for the numericalsolution fo the following singular integral equation arising from a problem of determining the distribution of stressin a thin elastic plate in the vicinity of a cruciform crack.These interpolants originate from the discretization of theintegral by two different quadrature formulas of interpolatorytype based on the zeros of Legendre orthogonal polynomials.The first quadrature is of product type and integrates exactlythe kernel; the second one is the well-known Gauss-Legendreformula. First we derive uniform convergence estimates for the two basicquadrature rules. Then by properly modifying the interpolantassociated with the Gauss-Legendre rule we prove its stabilityand derive for it a uniform error estimate of the type O(n^–4+),>0 as small as we like. We also show that if we had beenable to prove the stability of the first (modified) interpolantwe would have obtained a similar convergence estimate. Finally,for the Gauss-Legendre interpolant we prove that in any closedsubinterval [ 1] (0, 1] the rate of convergence is at leastO(n^–6+). Some numerical results which show the accuracy of our approximantsare also presented.     

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     

Optimal Recovery of the Derivative of Bounded Analytic Functions

Given (–1, 0), n N, we discuss the optimal recoveryof (), for analytic and bounded in < 1, from the knowledge of the values of at n points z1,.zm[0,l),where these points are chosen to produce the least possibleintrinsic error. The optimal algorithms are explicitly determined.     

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.     

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 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.     

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.     

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.     

A numerical method using the Prfer transformation for the calculation of eigenpairs of two-parameter Sturm-Liouville problems

A shooting method for coupled Prfer equations is discussedfor numerical solution of two-parameter Sturm-Liouville problems. Research supported in part by grants from the NSERC of Canada.     

Numerical simulations of measurements of capillary contact angles

In a recent paper, Fischer and Finn have proposed a procedureto improve the accuracy in the measurement of capillary contactangles, based on the use of vessels with canonical cross-sections.We simulate numerically the behaviour of such shapes for a numberof cross-sections and fluid contact angles. Our approximationconsists of the minimization of a suitable convex functionaldiscretized by finite elements. e-mail: bellettini{at}sns.it e-mail: paolini{at}isa.mat.unimi.it     

Finite-difference schemes for scalar conservation laws with source terms

Explicit and semi-implicit finite-difference schemes approximatingnon-homogeneous scalar conservation laws are analyzed. Optimalerror bounds independent of the stiffness of the underlyingequation are presented. This author has been supported by The Norwegian Research Council(NFR), program No 100284/431. e-mail: schroll{at}igpm.rwth-aachen.de This author has been supported by The Norwegian Research Council(NFR), program Nos 100284/431 and STP.29643. e-mail: ragnar{at}ifi.uio.no     

The Stability of the {theta}-methods in the Numerical Solution of Delay Differential Equations

This paper deals with the stability analysis of numerical methodsfor the solution of delay differential equations. We focus onthe behaviour of the one-leg -method and the linear -methodin the solution of the linear test equation U'(t)=U(t)+µU(t- ), with >0 and complex ,µ The stability regions forboth of these methods are determined. The regions turn out tobe equal to each other only if =0 or =1.     

The Method of Fundamental Solutions for the Solution of Nonlinear Plane Potential Problems

The method of fundamental solutions is described for the solutionof elliptic boundary value problems governed by Laplace's equationin the plane subject to nonlinear radiation-type boundary conditions.The effectiveness of the method is demonstrated by examiningits performance on two problems from the literature, and comparisonsare made with published results obtained using boundary elementmethods. Portions of this work were conducted while this author wasa visiting assistant professor at the University of Kentucky. Partially supported by the National Science Foundation undergrant MCS-8303287 and grant RII-8610671, and by the Commonwealthof Kentucky through the Kentucky EPSCoR Program.