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.     

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 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     

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 .     

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}.     

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.     

Rate of convergence of numerical approximations to homoclinic bifurcation points

For x=f (x, ), x Rⁿ, R, having a hyperbolic or semihyperbolicequilibrium p(), we study the numerical approximation of parametervalues ^* at which there is an orbit homoclinic to p(). We approximate^* by solving a finite-interval boundary value problem on J=[T_–,T₊], T_–<0<T₊, with boundary conditions that sayx(T_–) and x(T₊) are in approximations to appropriate invariantmanifolds of p(). A phase condition is also necessary to makethe solution unique. Using a lemma of Xiao-Biao Lin, we improve,for certain phase conditions, existing estimates on the rateof convergence of the computed homoclinic bifurcation parametervalue , to the true value ^*. The estimates we obtain agree withthe rates of convergence observed in numerical experiments.Unfortunately, the phase condition most commonly used in numericalwork is not covered by our results.     

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopytypes occurring in a k-parameter semi-algebraic family of setsin R, each defined in terms of m quadratic inequalities. Ourbound is exponential in k and m, but polynomial in . More precisely,we prove the following. Let R be a real closed field and let = {P₁, ... , P_m} R[Y₁, ... ,Y,X₁, ... ,X_k], with deg_Y(P_i) 2, deg_X(P_i) d, 1 i m. Let S R^+k be a semi-algebraic set,defined by a Boolean formula without negations, with atoms ofthe form P 0, P 0, P . Let : R^+k R^k be the projection onthe last k coordinates. Then the number of stable homotopy typesamongst the fibers S_x = ^–1(x) S is bounded by (2^mkd)^O(mk).     

Convergence of a fully discrete approximation for advected mean curvature flows

Let (t) be a closed curve in R² which propagates in its normaldirection n with velocity V = --q.n-g, where is the mean curvatureof (t) and g and q are given represent, respectively, a forcingterm and a vector field. In this paper we prove that such flowscan be approximated by numerical solutions of advection Allen-Cahnequations. It is shown that the zero level set of the fullydiscrete solution using explicit time stepping converges evenpast singularities to the true interface provided that no fatteningoccurs and , h² O(⁴), where h and denote the mesh size andthe time step. For smooth flows an optimal O(²)-rate of convergenceis derived provided , h² O(⁵). The analysis is based on constructingfully discrete barriers via an explicit parabolic projectionand Lipschitz dependence of the viscosity solutions with respectto perturbations of data.     

The Isolated Points of G{rho} and the w*-Strongly Exposed Points of P{rho}(G)0

Let G be a separable locally compact group and let be its dualspace with Fell's topology. It is well known that the set P(G)of continuous positive-definite functions on G can be identifiedwith the set of positive linear functionals on the group C^*-algebraC^*(G). We show that if is discrete in , then there exists anonzero positive-definite function associated with such that is a w^*-strongly exposed point of P(G)₀, where P(G)₀={f P(G):f(e)1. Conversely, if some nonzero positive-definite function associatedwith is a w^*-strongly exposed point of P(G)₀, then is isolatedin . Consequently, G is compact if and only if, for every ,there exists a nonzero positive-definite function associatedwith that is a w^*-strongly exposed point of P(G)₀. If, in addition,G is unimodular and , then is isolated in if and only if somenonzero positive-definite function associated with is a w^*-stronglyexposed point of P(G)₀, where is the left regular representationof G and is the reduced dual space of G. We prove that if B(G)has the Radon–Nikodym property, then the set of isolatedpoints of (so square-integrable if G is unimodular) is densein . It is also proved that if G is a separable SIN-group, thenG is amenable if and only if there exists a closed point in. In particular, for a countable discrete non-amenable groupG (for example the free group F₂ on two generators), there isno closed point in its reduced dual space .     

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)^–?.     

Geometry of Critical Loci

Let :(Z,z)(U,0) be the germ of a finite (that is, proper with finite fibres)complex analytic morphism from a complex analytic normal surfaceonto an open neighbourhood U of the origin 0 in the complexplane C². Let u and v be coordinates of C² defined on U. Weshall call the triple (, u, v) the initial data. Let stand for the discriminant locus of the germ , that is,the image by of the critical locus of . Let ()_A be the branches of the discriminant locus at O whichare not the coordinate axes. For each A, we define a rational number d by where I(–, –) denotes the intersection number at0 of complex analytic curves in C². The set of rational numbersd, for A, is a finite subset D of the set of rational numbersQ. We shall call D the set of discriminantal ratios of the initialdata (, u, v). The interesting situation is when one of thetwo coordinates (u, v) is tangent to some branch of , otherwiseD = {1}. The definition of D depends not only on the choiceof the two coordinates, but also on their ordering. In this paper we prove that the set D is a topological invariantof the initial data (, u, v) (in a sense that we shall definebelow) and we give several ways to compute it. These resultsare first steps in the understanding of the geometry of thediscriminant locus. We shall also see the relation with thegeometry of the critical locus.     

Central L-Values and Toric Periods for GL(2)

Let be a cusp form on GL(2) over a number field F and let Ebe a quadratic extension of F. Denote by _E the base change of to E and by a unitary character of A^x_E/ E^x. We use the relativetrace formula to give an explicit formula for L(1/2, _E ) interms of period integrals of Gross–Prasad test vectors.We give an application of this formula to equidistribution ofgeodesics on a hyperbolic 3-fold.     

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.     

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.     

On hearing the shape of a bounded domain with Robin boundary conditions

The asymptotic expansions of the trace of the heat kernel (t)= [sum ]_j=1 exp(-t_j) for small positive t, where {_j} _j=1 arethe eigenvalues of the negative Laplacian -_n = -[sum ]ⁿ_k=1 (/x^k)²in Rⁿ (n = 2 or 3), are studied for a general multiply connectedbounded domain which is surrounded by simply connected boundeddomains _i with smooth boundaries _i (i = 1,...,m), where smoothfunctions Y_i (i = 1,...,m) are assuming the Robin boundary conditions(n_i + Y_i) = 0 on _i. Here /n_i denote differentiations along theinward-pointing normals to _i (i = 1,...,m). Some applicationsof an ideal gas enclosed in the multiply connected bounded containerwith Neumann or Robin boundary conditions are given.     

Lp Lq Estimates for Parabolic Systems in Non-Divergence Form with VMO Coefficients

Consider a parabolic NxN-system of order m on ⁿ with top-ordercoefficients a VMOL. Let 1 < p, q < and let be a Muckenhouptweight. It is proved that systems of this kind possess a uniquesolution u satisfying whereAu = _||m a Du and J = [0,). In particular, choosing = 1, therealization of A in L^p(ⁿ)^N has maximal L^p – L^q regularity.     

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.     

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.