Calderon-Zygmund Operators on Mixed Lebesgue Spaces and Applications to Null Forms

The boundedness of Calderón–Zygmund operators isproved in the scale of the mixed Lebesgue spaces. As a consequence,the boundedness of the bilinear null forms Q_{i j} (u,) =_i u_j - _j u_i , Q₀(u,)=u_{t t} -_x u· _x on various space–timemixed Sobolev–Lebesgue spaces is shown.     

Superharmonic Extension from the Boundary of a Domain

A sharp geometric condition is given for a bounded domain tohave the property that for each superharmonic function u ona neighbourhood of , there is a superharmonic function on Rⁿsuch that = u on .     

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.     

On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smoothfunction on M. Let p^h(x) = inf||–₁(Ric_x(,)–2Hess(h_x(,)).Here Ric_x denotes the Ricci curvature at x and Hess(h) is theHessian of h. Then M has finite fundamental group if ^h–p^h<0. Here ^h =:+2L_h is the Bismut-Witten Laplacian. This leadsto a quick proof of recent results on extension of Myers' theoremto manifolds with mostly positive curvature. There is also asimilar result for noncompact manifolds.     

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     

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.     

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.     

BOUNDARY CONCENTRATION IN RADIAL SOLUTIONS TO A SYSTEM OF SEMILINEAR ELLIPTIC EQUATIONS

We study concentration phenomena for the system in the unit ball B₁ of ³ with Dirichlet boundaryconditions. Here , , > 0 and p > 1. We prove the existenceof positive radial solutions (, ) such that concentrates ata distance (/2)|log | away from the boundary B₁ as the parameter tends to 0. The approach is based on a combination of Lyapunov–Schmidtreduction procedure together with a variational method.     

The P-version Penalty Finite Element Method

A p-version penalty finite element method is used to solve themodel problem –u=f in , u=g on . Error estimates are derivedin H¹-norm. The p-version penalty method with extrapolationyields an approximate solution which converges at the optimalrate. Numerical results show the effectiveness of the p-versionpenalty method with extrapolation.     

Higher Generation Subgroup Sets and the Virtual Cohomological Dimension of Graph Products of Finite Groups

We introduce panels of stabilizer schemes (K, G_*) associatedwith finite intersection-closed subgroup sets of a given groupG, generalizing in some sense Davis' notion of a panel structureon a triangulated manifold for Coxeter groups. Given (K, G_*),we construct a G-complex X with K as a strong fundamental domainand simplex stabilizers conjugate to subgroups in . It turnsout that higher generation properties of in the sense of Abels-Holzare reflected in connectivity properties of X. Given a finite simplicial graph and a non-trivial group G()for every vertex of , the graph product G() is the quotientof the free product of all vertex groups modulo the normal closureof all commutators [G(), G(w)] for which the vertices , w areadjacent. Our main result allows the computation of the virtualcohomological dimension of a graph product with finite vertexgroups in terms of connectivity properties of the underlyinggraph .     

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.     

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.     

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)^–?.     

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}.     

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.     

Weighted Integral Inequalities for the Hardy Type Operator and the FractionalMaximal Operator

Let w(x), u(x) and (x) be weight functions. In this paper, underappropriate conditions on Young's functions ₁, ₂ we characterizethe inequality for the Hardy-typeoperator T defined in [1] and the inequality for the fractional maximal operator M_{, ;} definedin [8], as well as the corresponding weak-type inequalities.     

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.     

Finite-Element Approximation of Elliptic Equations with a Neumann or Robin Condition on a Curved Boundary

This paper considers a finite-element approximation of a second-orderself adjoint elliptic equation in a region Rⁿ (with n=2 or 3)having a curved boundary on which a Neumann or Robin conditionis prescribed. If the finite-element space defined over , a union of elements, has approximation power h^kin the L² norm, and if the region of integration is approximatedby ^h with dist (, ^h)Ch^k, then it is shown that one retains optimalrates of convergence for the error in the H¹ and L² norms, whetherQ^h is fitted or unfitted , provided that the numerical integration scheme has sufficientaccuracy.