A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems

^** Email: emmanuil.georgoulis{at}mcs.le.ac.uk^*** Email: al{at}maths.strath.ac.uk We consider a variant of the hp-version interior penalty discontinuousGalerkin finite element method (IP-DGFEM) for second-order problemsof degenerate type. We do not assume uniform ellipticity ofthe diffusion tensor. Moreover, diffusion tensors of arbitraryform are covered in the theory presented. A new, refined recipefor the choice of the discontinuity-penalization parameter (thatis present in the formulation of the IP-DGFEM) is given. Makinguse of the recently introduced augmented Sobolev space framework,we prove an hp-optimal error bound in the energy norm and anh-optimal and slightly p-suboptimal (by only half an order ofp) bound in the L² norm (the latter, for the symmetric versionof the IP-DGFEM), provided that the solution belongs to an augmentedSobolev space.     

Gleason Property and Extensions of States on Projection Logics

We prove that every state on the projection logic P(M) of avon Neumann algebra M not containing a direct summand of typeI₂ extends to a state of an arbitrary larger unital logic L.We also show that if a C^*-algebra enjoys the Gleason property,and if it possesses sufficiently many projections, then an analogousresult can be derived. Moreover, we prove that the extensionscan be taken linear in a complete order unit norm space associatedwith L. (Results of this paper generalize results of [22] andmay contribute to the noncommutative measure theory, convextheory of state spaces and foundations of quantum physics.)     

Smoothness Properties of the Unit Ball in a JB*-Triple

An element a of norm one in a JB^*-triple A is said to be smoothif there exists a unique element x in the unit ball A₁^* of thedual A^* of A at which a attains its norm, and is said to beFréchet-smooth if, in addition, any sequence (x_n) ofelements in A₁^* for which (x_n(a)) converges to one necessarilyconverges in norm to x. The sequence (a²ⁿ⁺¹) of odd powers ofa converges in the weak^*-topology to a tripotent u(a) in theJBW^*-envelope A^** of A. It is shown that a is smooth if andonly if u(a) is a minimal tripotent in A^** and a is Fréchet-smoothif and only if, in addition, u(a) lies in A.     

Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows

Endre Süli ^{We develop a posteriori upper and lower error bounds for mixedfinite-element approximations of a general family of steady,viscous, incompressible quasi-Newtonian flows in a bounded Lipschitzdomain ; thefamily includes degenerate models such as the power law model,as well as non-degenerate ones such as the Carreau model. Theunified theoretical framework developed herein yields residual-baseda posteriori bounds which measure the error in the approximationof the velocity in the W1, r() norm and that of the pressurein the Lr'() norm, 1/r + 1/r' = 1, r (1, ).}     

Local Rigidity of Infinite-Dimensional Teichmuller Spaces

This paper presents a rigidity theorem for infinite-dimensionalBergman spaces of hyperbolic Riemann surfaces, which statesthat the Bergman space A¹(M), for such a Riemann surface M,is isomorphic to the Banach space of summable sequence, l¹.This implies that whenever M and N are Riemann surfaces thatare not analytically finite, and in particular are not necessarilyhomeomorphic, then A¹(M) is isomorphic to A¹(N). It is knownfrom V. Markovic that if there is a linear isometry betweenA¹(M) and A¹(N), for two Riemann surfaces M and N of non-exceptionaltype, then this isometry is induced by a conformal mapping betweenM and N. As a corollary to this rigidity theorem presented here,taking the Banach duals of A¹(M) and l¹ shows that the spaceof holomorphic quadratic differentials on M, Q(M), is isomorphicto the Banach space of bounded sequences, l. As a consequenceof this theorem and the Bers embedding, the Teichmüllerspaces of such Riemann surfaces are locally bi-Lipschitz equivalent.     

On the Sloan iteration applied to integral equations of the first kind

When the piecewise constant collocation method is used to solvean integral equation of the first kind with logarithmic kernel,the convergence rate is O(h) in the L₂ norm. In this note weshow that O(h³) or O(h⁵) convergence in any Sobolev norm (andthus, for example, in L) may be obtained by a simple cheap postprocessingof the original collocation solution. The construction of thepostprocessor is based on writing the first kind equation asa second kind equation, and applying the Sloan iteration tothe latter equation. The theoretical convergence rates are verifiedin a numerical example.     

Best Constants for Uncentred Maximal Functions

We precisely evaluate the operator norm of the uncentred Hardy–Littlewoodmaximal function on L^p(R¹). Consequently, we compute the operatornorm of the ‘strong’ maximal function on L^p(Rⁿ),and we observe that the operator norm of the uncentred Hardy–Littlewoodmaximal function over balls on L^p(Rⁿ) grows exponentially asn. 1991 Mathematics Subject Classification 42B25.     

On The Greatest Prime Factor of axm+byn, II

We provide a new lower bound for the greatest prime factor ofthe norm of algebraic numbers of the form ax^m+byⁿ. The improvementconcerns the dependence on the number field containing a, b,x and y. 1991 Mathematics Subject Classification 11D61.     

Improving the accuracy of the mini-element approximation to Navier-Stokes equations

^** Email: blanca{at}imati.cnr.it^*** Email: frutos{at}mac.cie.uva.es^**** Corresponding author. Email: julia.novo{at}uam.es A technique to improve the accuracy of the mini-element approximationto incompressible the Navier–Stokes equations is introduced.Once the mini-element approximation has been computed at a fixedtime, the linear part of this approximation is postprocessedby solving a discrete Stokes problem. The bubble functions neededto stabilize the approximation to the Navier–Stokes equationsare not used at the postprocessing step. This postprocessingprocedure allows us to increase by one unit (up to a logarithmicterm) the H¹ norm rate of convergence of the velocity and correspondinglythe L² norm of the pressure. An error analysis of the algorithmis performed.     

On Hardy, BMO and Lipschitz spaces of invariantly harmonic functions in the unit ball

The invariantly harmonic functions in the unit ball Bⁿ in Cⁿare those annihilated by the Bergman Laplacian . The Poisson-Szegökernel P(z,) solves the Dirichlet problem for : if f C(Sⁿ),the Poisson-Szegö transform of f, where d is the normalized Lebesgue measure on Sⁿ,is the unique invariantly harmonic function u in Bⁿ, continuousup to the boundary, such that u=f on Sⁿ. The Poisson-Szegötransform establishes, loosely speaking, a one-to-one correspondencebetween function theory in Sⁿ and invariantly harmonic functiontheory in Bⁿ. When n 2, it is natural to consider on Sⁿ functionspaces related to its natural non-isotropic metric, for theseare the spaces arising from complex analysis. In the paper,different characterizations of such spaces of smooth functionsare given in terms of their invariantly harmonic extensions,using maximal functions and area integrals, as in the correspondingEuclidean theory. Particular attention is given to characterizationin terms of purely radial or purely tangential derivatives.The smoothness is measured in two different scales: that ofSobolev spaces and that of Lipschitz spaces, including BMO andBesov spaces. 1991 Mathematics Subject Classification: 32A35,32A37, 32M15, 42B25.     

Multiquadratic Extensions, Rigid Fields and Pythagorean Fields

Let F be a field of characteristic other than 2. Let F⁽²⁾ denotethe compositum over F of all quadratic extensions of F, letF⁽³⁾ denote the compositum over F⁽²⁾ of all quadratic extensionsof F⁽²⁾ that are Galois over F, and let F^{3} denote the compositumover F⁽²⁾ of all quadratic extensions of F⁽²⁾. This paper showsthat F⁽³⁾ = F^{3} if and only if F is a rigid field, and thatF⁽³⁾ = K⁽³⁾ for some extension K of F if and only if F is Pythagoreanand . The proofs depend mainly on the behavior of quadratic forms over quadratic extensions,and the corresponding norm maps.     

Approximation Numbers of Sobolev Embedding Operators on an Interval

Consider the Sobolev embedding operator from the space of functionsin W^1,p(I) with average zero into L^p, where I is a finite intervaland p>1. This operator plays an important role in recentwork. The operator norm and its approximation numbers in closedform are calculated. The closed form of the norm and approximationnumbers of several similar Sobolev embedding operators on afinite interval have recently been found. It is proved in thepaper that most of these operator norms and approximation numberson a finite interval are the same.     

Updating and downdating an upper trapezoidal sparse orthogonal factorization{dagger}

^** Email: santos{at}ctima.uma.es^*** Corresponding Author. Email: pablito{at}ctima.uma.es We describe how to update and downdate an upper trapezoidalsparse orthogonal factorization, namely the sparse QR factorizationof A_k^T, where A_k is a ‘tall and thin’ full columnrank matrix formed with a subset of the columns of a fixed matrixA. In order to do this, we have adapted Saunders' techniquesof the early 1970s for square matrices, to rectangular matrices(with fewer columns than rows) by using the static data structureof George and Heath of the early 1980s but allowing row downdatingon it. An implicitly determined column permutation allows usto dispense with computing a new ordering after each update/downdate;it fits well into the LINPACK downdating algorithm and ensuresthat the updated trapezoidal factor will remain sparse. We giveall the necessary formulae even if the orthogonal factor isnot available, and we comment on our implementation using thesparse toolbox of MATLAB 5.     

Convergence rates for the coupling of FEM and collocation BEM

We prove convergence of the coupling of finite and boundaryelements where Galerkin's methd is used for finite elementsand collocation for boundary elements. We consider linear ellipticboundary value problems in two dimensions, in particular problemsin elasticity. The mesh width k of the boundary elements andthe mesh width h of the finite elements are required to satisfykßh with suitable ß. Asymptotic error estimatesin the energy norm and in the L²-norm are derived. Numericalexamples are included.     

Invariant Subspaces of Bergman Spaces on Slit Domains

In this paper, we characterize the z-invariant subspaces thatlie between the Bergman spaces A^p(G) and A^p(G\K), where 1 <p < , G is a bounded region in C, and K is a closed subsetof a simple compact C¹ arc.     

A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces

An analogue of the Paley–Wiener theorem is developed forweighted Bergman spaces of analytic functions in the upper half-plane.The result is applied to show that the invariant subspaces ofthe shift operator on the standard Bergman space of the unitdisk can be identified with those of a convolution Volterraoperator on the space L²(⁺, (1/t)dt).     

Sharp Geometric Poincare Inequalities for Vector Fields and Non-Doubling Measures

We derive Sobolev–Poincaré inequalities that estimatethe L^q(d µ) norm of a function on a metric ball when µis an arbitrary Borel measure. The estimate is in terms of theL¹(d ) norm on the ball of a vector field gradient of the function,where d dx is a power of a fractional maximal function of µ.We show that the estimates are sharp in several senses, andwe derive isoperimetric inequalities as corollaries. 1991 MathematicsSubject Classification: 46E35, 42B25.     

Torsional Rigidity of Minimal Submanifolds

We prove explicit upper bounds for the torsional rigidity ofextrinsic domains of minimal submanifolds P^m in ambient Riemannianmanifolds Nⁿ with a pole p. The upper bounds are given in termsof the torsional rigidities of corresponding Schwarz symmetrizationsof the domains in warped product model spaces. Our main resultsare obtained via previously established isoperimetric inequalities,which are here extended to hold for this more general settingbased on warped product comparison spaces. We also characterizethe geometry of those situations in which the upper bounds forthe torsional rigidity are actually attained and give conditionsunder which the geometric average of the stochastic mean exittime for Brownian motion at infinity is finite. 2000 MathematicsSubject Classification 53C42, 58J65, 35J25, 60J65.     

Convergence of a Finite-Difference Scheme for Second-Order Hyperbolic Equations with Variable Coefficients

We study the convergence of a finite-difference scheme for thefirst initial-boundary-value problem for linear second-orderhyperbolic equations with variable coefficients. Using the bilinearversion of the Bramble-Hilbert lemma we prove that the convergencerate in the discrete energy norm is of the order h^–2 ifthe exact solution belongs to the Sobolev space W₂(Q) with 2<<4.