Mapped Finite Elements on Hexahedra. Necessary and Sufficient Conditions for Optimal Interpolation Errors

This paper considers finite elements which are defined on hexahedral cells via a reference transformation which is in general trilinear. For affine reference mappings, the necessary and sufficient condition for an interpolation order O(h ^k+1) in the L ²-norm and O(h ^k ) in the H ¹-norm is that the finite dimensional function space on the reference cell contains all polynomials of degree less than or equal to k. The situation changes in the case of a general trilinear reference transformation. We will show that on general meshes the necessary and sufficient condition for an optimal order for the interpolation error is that the space of polynomials of degree less than or equal to k in each variable separately is contained in the function space on the reference cell. Furthermore, we will show that this condition can be weakened on special families of meshes. These families which are obtained by applying usual refinement techniques can be characterized by the asymptotic behaviour of the semi-norms of the reference mapping.     

Approximation properties of the Generalized Finite Element Method

In this paper, we have obtained an approximation result in the Generalized Finite Element Method (GFEM) that reflects the global approximation property of the Partition of Unity (PU) as well as the approximability of the local approximation spaces. We have considered a GFEM, where the underlying PU functions reproduce polynomials of degree l. With the space of polynomials of degree k serving as the local approximation spaces of the GFEM, we have shown, in particular, that the energy norm of the GFEM approximation error of a smooth function is O(h ^l + k ). This result cannot be obtained from the classical approximation result of GFEM, which does not reflect the global approximation property of the PU.     

Some researches on multivariate Lagrange interpolation along the sufficiently intersected algebraic manifold

In this article, Lagrange interpolation by polynomials in several variables is studied. Particularly on the sufficiently intersected algebraic manifolds, we discuss the dimension about the interpolation space of polynomials. After defining properly posed set of nodes (or PPSN for short) along the sufficiently intersected algebraic manifolds, we prove the existence of PPSN and give the number of points in PPSN of any degree. Moreover, in order to compute the number of points in PPSN concretely, we propose the operator ? _k with reciprocal difference.     

Polynomial interpolation of operators

Necessary and sufficient conditions for the solvability of the polynomial operator interpolation problem in an arbitrary vector space are obtained (for the existence of a Hermite-type operator polynomial, conditions are obtained in a Hilbert space). Interpolational operator formulas describing the whole set of interpolants in these spaces as well as a subset of those polynomials preserving operator polynomials of the corresponding degree are constructed. In the metric of a measure space of operators, an accuracy estimate is obtained and a theorem on the convergence of interpolational operator processes is proved for polynomial operators. Applications of the operator interpolation to the solution of some problems are described. Bibliography: 134 titles. This paper is a continuation of the work published inObchyslyuval'na ta Prykladna Maternatyka, No. 78 (1994). The numeration of chapters, assertions, and formulas is continued. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 79, 1995, pp 10–116.     

Comparison of a mixed least-squares formulation using different approximation spaces

The main goal of the present work is the comparison of the performance of a least-squares mixed finite element formulation where the solution variables (displacements and stresses) are interpolated using different approximation spaces. Basis for the formulation is a weak form resulting from the minimization of a least-squares functional, compare e.g. [1]. As suitable functions for standard interpolation polynomials of Lagrangian type are chosen. For the conforming discretization of the Sobolev space vector-valued Raviart-Thomas interpolation functions, see also [2], are used. The resulting elements are named as P_mP_k and RT_mP_k. Here m (stresses) and k (displacements) denote the approximation order of the particular interpolation function. For the comparison we consider a two-dimensional cantilever beam under plain strain conditions and small strain assumptions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Structure of the Taylor Series in Clifford and Quaternionic Analysis

The Fueter variables form a basis of the space of (quaternionic or Cliffordian) hyperholomorphic homogeneous polynomials of degree one, and their symmetrized products give the respective bases of spaces of hyperholomorphic homogeneous polynomials for any degree k. In the present paper we introduce new bases, i.e., new types of hyperholomorphic variables which lead to the Taylor-type series expansions reflecting the structure of the set of all (quaternionic or Cliffordian algebra-valued) hyperholomorphic functions.     

On the Relation between Piecewise Polynomial and Rational Approximation in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Bold">R</Emphasis><Superscript>2</Superscript>)

In the univariate case there are certain equivalences between the nonlinear approximation methods that use piecewise polynomials and those that use rational functions. It is known that for certain parameters the respective approximation spaces are identical and may be described as Besov spaces. The characterization of the approximation spaces of the multivariate nonlinear approximation by piecewise polynomials and by rational functions is not known. In this work we compare between the two methods in the bivariate case. We show some relations between the approximation spaces of piecewise polynomials defined on n triangles and those of bivariate rational functions of total degree n which are described by n parameters. Thus we compare two classes of approximants with the same number Cn of parameters. We consider this the proper comparison between the two methods.     

Divided Differences and Linearly Recursive Sequences

We show that the theory of divided differences is a natural tool for the study of linearly recurrent sequences. The divided differences functional associated with a monic polynomial w on degree n + 1 yields a vector space isomorphism between the space of polynomials of degree at most equal to n and the space of linearly recurrent sequences f that satisfy the difference equation w(E)f=0 where E is the usual shift operator. Using such isomorphisms, we can translate problems about recurrent sequences into simple problems about polynomials. We present here a new approach to the theory of divided differences, using only generating functions and elementary linear algebra, which clarifies the connections of divided differences with rational functions, polynomial interpolation, residues, and partial fractions decompositions.     

Partial differential equation approach to F 4

Singular vectors of a representation of a finite-dimensional simple Lie algebra are weight vectors in the underlying module that are nullified by positive root vectors. In this article, we use partial differential equations to explicitly find all the singular vectors of the polynomial representation of the simple Lie algebra of type F ₄ over its 26-dimensional basic irreducible module, which also supplements a proof of the completeness of Brion’s abstractly described generators. Moreover, we show that the number of irreducible submodules contained in the space of homogeneous harmonic polynomials with degree k ⩾ 2 is greater than or equal to 〚k/3〛 + 〚(k − 2)/3〛 + 2.     

Decomposable symmetric mappings between infinite-dimensional spaces

Decomposable mappings from the space of symmetric k-fold tensors over E, , to the space of k-fold tensors over F, , are those linear operators which map nonzero decomposable elements to nonzero decomposable elements. We prove that any decomposable mapping is induced by an injective linear operator between the spaces on which the tensors are defined. Moreover, if the decomposable mapping belongs to a given operator ideal, then so does its inducing operator. This result allows us to classify injective linear operators between spaces of homogeneous approximable polynomials and between spaces of nuclear polynomials which map rank-1 polynomials to rank-1 polynomials.     

Interpolation and Embeddings of Weighted Tent Spaces

Given a metric measure space X, we consider a scale of function spaces \(T^{p,q}_s(X)\), called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on X we identify some associated interpolation spaces, in particular certain real interpolation spaces. These are identified with a new scale of function spaces, which we call Z -spaces, that have recently appeared in the work of Barton and Mayboroda on elliptic boundary value problems with boundary data in Besov spaces. We also prove Hardy–Littlewood–Sobolev-type embeddings between weighted tent spaces.     

Methoden zur Bestimmung von Reprokernen

Methods to determine reproducing kernels. The explicit representation of continuous linear functionals on a Hilbert space by reprokernels is significant for interpolation and approximation. Starting with the kernels theorem, due to Schwartz, we develop methods to determine reprokernels for the Sobolev spaces W₂^k(Ω) if Ω R¹, and for some subspaces of W₂^k(Ω) if ΩRⁿ. Then we determine reprokernels for tensor products of Hilbert spaces. In addition to this we consider three types of limits of reprokernels.     

On Romanovski–Jacobi polynomials and their related approximation results

The aim of this article is to present the essential properties of a finite class of orthogonal polynomials related to the probability density function of the F -distribution over the positive real line. We introduce some basic properties of the Romanovski–Jacobi polynomials, the Romanovski–Jacobi–Gauss type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kinds of finite orthogonal polynomials and other classes of infinite orthogonal polynomials. Moreover, we derive spectral Galerkin schemes based on a Romanovski–Jacobi expansion in space and time to solve the Cauchy problem for a scalar linear hyperbolic equation in one and two space dimensions posed in the positive real line. Two numerical examples demonstrate the robustness and accuracy of the schemes.     

Centralisers of spaces of symmetric tensor products and applications

We show that the centraliser of the space of n-fold symmetric injective tensors, n≥2, on a real Banach space is trivial. With a geometric condition on the set of extreme points of its dual, the space of integral polynomials we obtain the same result for complex Banach spaces. We give some applications of this results to centralisers of spaces of homogeneous polynomials and complex Banach spaces. In addition, we derive a Banach-Stone Theorem for spaces of vector-valued approximable polynomials. This project was supported in part by Enterprise Ireland, International Collaboration Grant – 2004 (IC/2004/009). The second author was also partially supported by PIP 5272,UBACYTX108 and PICT 03-15033     

The Density of Translates of Zonal Kernels on Compact Homogeneous Spaces

Compact manifolds embedded in Euclidean space which have a transitive group G of linear isometries, such as the spheres with the rotation group or the “flat” tori with the group of rotations in each coordinate direction, admit a natural notion of a continuous G-invariant kernel function k(x, y), which generalizes the idea of a radial or distance-dependent function on the spheres and tori. In connection with a study of quasi-interpolation on these spaces, we have reproved and extended results of Sun for the spheres to characterize those kernels for which the span of the translates, ∑ a_nk(x, y_n), is dense in the continuous functions. The essence of the characterization is that the integral operator with G-invariant kernel k(x, y) must be non-singular when restricted to the space of nth degree polynomial functions. This requires that the polynomials be invariant under all such linear operators, which is true for many compact homogeneous M including the spheres, tori, and others. In fact the non-singularity must hold only on any finite-dimensional space of zonal polynomials, those which are pointwise fixed by the subgroup of all isometries fixing a single point on M. In practical terms this later condition is verified by choosing one point on the manifold (the north pole on the spheres or the identity element on the flat tori), picking some basis for the polynomials of given degree which are fixed under the isometries leaving the pole invariant, and testing whether the integral operator (which leaves this space invariant) has a non-singular matrix. In all the cases considered, where the family of G-invariant kernels lead to commuting operator families, there are diagonalizing bases for this restricted operator, and the characterization becomes the non-vanishing of the appropriate Fourier-like coefficients.     

Hypergraphical Codes Arising from Binary Trades

By considering the null space of incidence matrices of trivial designs over GF(2) (the space of 1- (v,k) trades overGF(2)) we obtain families of codes which are optimal for some v and k. Moreover, by generalizing the concept of bond space, the weight enumerator polynomials for these codes are obtained.     

Ultraconvergence of the derivative of high-order finite element method for elliptic problems with constant coefficients

In this article, for second order elliptic problems with constant coefficients, the local ultraconvergence of the derivative of finite element method using piecewise polynomials of degrees k (k ≥ 2) is studied by the interpolation postprocessing technique. Under suitable regularity and mesh conditions, we prove that at an interior vertex, which is away from the boundary with a fixed distance, the gradient of the postprecessed finite element solution using piecewise polynomials of degrees k (k ≥ 2) converges to the gradient of the exact solution with order . Numerical experiments are used to illustrate our theoretical findings.     

On holomorphic functions attaining their norms

We show that on a complex Banach space X, the functions uniformly continuous on the closed unit ball and holomorphic on the open unit ball that attain their norms are dense provided that X has the Radon-Nikodym property. We also show that the same result holds for Banach spaces having a strengthened version of the approximation property but considering just functions which are also weakly uniformly continuous on the unit ball. We prove that there exists a polynomial such that for any fixed positive integer k, it cannot be approximated by norm attaining polynomials with degree less than k. For , a predual of a Lorentz sequence space, we prove that the product of two polynomials with degree less than or equal two attains its norm if, and only if, each polynomial attains its norm.     

On Harmonic Weight Enumerators of Binary Codes

We define some new polynomials associated to a linear binary code and a harmonic function of degree k. The case k=0 is the usual weight enumerator of the code. When divided by (xy) ^k , they satisfy a MacWilliams type equality. When applied to certain harmonic functions constructed from Hahn polynomials, they can compute some information on the intersection numbers of the code. As an application, we classify the extremal even formally self-dual codes of length 12.