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.     

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.     

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.     

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     

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.     

A note on least-squares mixed finite elements in relation to standard and mixed finite elements

^** Email: brandts{at}science.uva.nl The least-squares mixed finite-element method for second-orderelliptic problems yields an approximation u_h V_h H₀¹() of thepotential u together with an approximation p_{h h} H(div ; )of the vector field p = – Au. Comparing u_h with the standardfinite-element approximation of u in V_h, and p_h with the mixedfinite-element approximation of p, it turns out that they arehigher-order perturbations of each other. In other words, theyare ‘superclose’. Refined a priori bounds and superconvergenceresults can now be proved. Also, the local mass conservationerror is of higher order than could be concluded from the standarda priori analysis.     

Boundaries of reduced free group C*-algebras

We prove that the crossed product C*-algebra C*_r(, ) of a freegroup with its boundary sits naturally between the reducedgroup C*-algebra C*_r and its injective envelope I(C*_r). In otherwords, we have natural inclusion C*_r C*_r(, ) I(C*_r) of C*-algebras.     

Finite-element wavelets on manifolds

We construct locally supported, continuous wavelets on manifolds that are given as the closure of a disjoint union of generalsmooth parametric images of an n-simplex. The wavelets are provento generate Riesz bases for Sobolev spaces H^s () when s (–1,3/2), if not limited by the global smoothness of . These resultsgeneralize the findings from Dahmen & Stevenson (1999) SIAMJ. Numer. Anal., 37, 319–352, where it was assumed thateach parametrization has a constant Jacobian determinant. Thewavelets can be arranged to satisfy the cancellation propertyof, in principle, any order, except for wavelets with supportsthat extend to different patches, which generally satisfy thecancellation property of only order 1.     

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.     

Anisotropic interpolation and quasi-Wilson element for narrow quadrilateral meshes

In this paper an anisotropic interpolation theorem is presentedthat can be easily used to check the anisotropy of an element.A kind of quasi-Wilson element is considered for second-orderproblems on narrow quadrilateral meshes for which the usualregularity condition _K/h_K c₀ > 0 is not satisfied, whereh_K is the diameter of the element K and _K is the radius of thelargest inscribed circle in K. Anisotropic error estimates ofthe interpolation error and the consistency error in the energynorm and the L²-norm are given. Furthermore, we give a Poincaréinequality on a trapezoid which improves a result of eniek.     

Finite elements on evolving surfaces

^** Email: gerd.dziuk{at}mathematik.uni-freiburg.de In this article, we define a new evolving surface finite-elementmethod for numerically approximating partial differential equationson hypersurfaces (t) in ⁿ⁺¹ which evolve with time. The keyidea is based on approximating (t) by an evolving interpolatedpolyhedral (polygonal if n = 1) surface _h(t) consisting of aunion of simplices (triangles for n = 2) whose vertices lieon (t). A finite-element space of functions is then definedby taking the set of all continuous functions on _h(t) whichare linear affine on each simplex. The finite-element nodalbasis functions enjoy a transport property which simplifiesthe computation. We formulate a conservation law for a scalarquantity on (t) and, in the case of a diffusive flux, derivea transport and diffusion equation which takes into accountthe tangential velocity and the local stretching of the surface.Using surface gradients to define weak forms of elliptic operatorsnaturally generates weak formulations of elliptic and parabolicequations on (t). Our finite-element method is applied to theweak form of the conservation equation. The computations ofthe mass and element stiffness matrices are simple and straightforward.Error bounds are derived in the case of semi-discretizationin space. Numerical experiments are described which indicatethe order of convergence and also the power of the method. Wedescribe how this framework may be employed in applications.     

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 .     

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.     

Gauge Fixing for Logarithmic Connections Over Curves and the Riemann-Hilbert Problem

Let be a compact Riemann surface of genus g, X={x₁, ..., x_n} a finite set of points, and ¹(log X) be the sheaf of 1-forms,holomorphic over \X and generated near x_j by dz_j/z_j for a coordinatez_j centred at x_j.     

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)     

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.     

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.     

Identity Theorems for Functions of Bounded Characteristic

Suppose that f(z) is a meromorphic function of bounded characteristicin the unit disk :|z|<1. Then we shall say that f(z)N. Itfollows (for example from [3, Lemma 6.7, p. 174 and the following])that where h₁(z), h₂(z) are holomorphic in and have positive realpart there, while ₁(z), ₂(z) are Blaschke products, that is, where p is a positive integer or zero, 0<|a_j|<1, c isa constant and (1–|a_j|)<. We note in particular that, if c0, so that f(z)0, (1.1) so that f(z)=0 only at the points a_j. Suppose now that z_j isa sequence of distinct points in such that |z_j|1 as j and (1–|z_j|)=. (1.2) If f(z_j)=0 for each j and fN, then f(z)0. N. Danikas [1] has shown that the same conclusion obtains iff(z_j)0 sufficiently rapidly as j. Let _j, _j be sequences of positivenumbers such that _j< and _j as j. Danikas then defines and proves Theorem A.     

A necessary and sufficient algebraic condition for the controllability of a thermoelastic plate equation

In this paper we give a necessary and sufficient algebraic conditionfor the approximate controllability of the following thermoelasticplate equation with Dirichlet boundary conditions w_tt + ²w + = a₁(x)u₁ + ... + a_m(x)u_m, t 0, x , _t – ß – w_t = d₁(x)u₁ + ... + d_m(x)u_m,t 0, x , = w = w = 0, t 0, x , where 0, ß > 0, is a sufficiently regular boundeddomain in R^N, a_i, d_i, L² (; R), the control functions u_i L²(0, t₁; R); i = 1, 2, ..., m. This condition is easy to checkand is given by Rank [P_jBA_jP_jBA²_jP_jB ... A^3_j–1_jP_jB] = 3_j,BU=b₁U₁+...+b_mU_m,b_i=[0, a_i, d_i], A_j=[0, –²_j, 0, 1, 0, –_j, 0, _j, –ß_j]P_j, j1, where _j, S are the eigenvalues of – with Dirichlet boundarycondition and P_j, S are the projections on the correspondingeigenspace.     

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)^–?.