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.     

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.     

Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient

We analyse approximate solutions generated by an upwind differencescheme (of Engquist–Osher type) for nonlinear degenerateparabolic convection–diffusion equations where the nonlinearconvective flux function has a discontinuous coefficient (x)and the diffusion function A(u) is allowed to be strongly degenerate(the pure hyperbolic case is included in our setup). The mainproblem is obtaining a uniform bound on the total variationof the difference approximation u, which is a manifestationof resonance. To circumvent this analytical problem, we constructa singular mapping (, ·) such that the total variationof the transformed variable z = (, u) can be bounded uniformlyin . This establishes strong L¹ compactness of z and, since(, ·) is invertible, also u. Our singular mapping isnovel in that it incorporates a contribution from the diffusionfunction A(u). We then show that the limit of a converging sequenceof difference approximations is a weak solution as well as satisfyinga Krukov-type entropy inequality. We prove that the diffusionfunction A(u) is Hölder continuous, implying that the constructedweak solution u is continuous in those regions where the diffusionis nondegenerate. Finally, some numerical experiments are presentedand discussed.     

Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients

^** Email: todor{at}math.ethz.ch^*** Corresponding author. Email: schwab{at}math.ethz.ch A scalar, elliptic boundary-value problem in divergence formwith stochastic diffusion coefficient a(x, ) in a bounded domainD ^d is reformulated as a deterministic, infinite-dimensional,parametric problem by separation of deterministic (x D) andstochastic ( ) variables in a(x, ) via Karhúnen–Loèveor Legendre expansions of the diffusion coefficient. Deterministic,approximate solvers are obtained by projection of this probleminto a product probability space of finite dimension M and sparsediscretizations of the resulting M-dimensional parametric problem.Both Galerkin and collocation approximations are considered.Under regularity assumptions on the fluctuation of a(x, ) inthe deterministic variable x, the convergence rate of the deterministicsolution algorithm is analysed in terms of the number N of deterministicproblems to be solved as both the chaos dimension M and themultiresolution level of the sparse discretization resp. thepolynomial degree of the chaos expansion increase simultaneously.     

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.     

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)^–?.     

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.     

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.