A fast algorithm for Gaussian type quadrature formulae with mixed boundary conditions and some lumped mass spectral approximations

After studying Gaussian type quadrature formulae with mixed boundary conditions, we suggest a fast algorithm for computing their nodes and weights. It is shown that the latter are computed in the same manner as in the theory of the classical Gauss quadrature formulae. In fact, all nodes and weights are again computed as eigenvalues and eigenvectors of a real symmetric tridiagonal matrix. Hence, we can adapt existing procedures for generating such quadrature formulae. Comparative results with various methods now in use are given. In the second part of this paper, new algorithms for spectral approximations for second-order elliptic problems are derived. The key to the efficiency of our algorithms is to find an appropriate spectral approximation by using the most accurate quadrature formula, which takes the boundary conditions into account in such a way that the resulting discrete system has a diagonal mass matrix. Hence, our algorithms can be used to introduce explicit resolutions for the time-dependent problems. This is the so-called lumped mass method. The performance of the approach is illustrated with several numerical examples in one and two space dimensions.

     

Negative Definite Cubature Formulae, Extremality and Delaunay Triangulation

Let Ω⊂ℝ ^d be a compact convex polytope of positive measure. We study cubature formulae on Ω which approximate the integral of every convex function f∈C(Ω) from above. They are called negative definite formulae or nd-formulae for short. In particular, we characterize nd-formulae by certain partitions of unity or, alternatively, by a class of positive linear operators. For aiming at ‘good’ nd-formulae, we introduce three extremal properties named as minimal, best and optimal. We show that the Delaunay triangulation and one of its generalizations give access to efficient algorithms for computing nd-formulae with one of these properties.     

Méthode d'homogénéisation auto-cohérente en calcul à la ruptureSelf-consistent homogeneization method in yield design

This work is aimed towards determining the macroscopic strength criterion of a heterogeneous material with a random microstructure. We use the self-consistent concept by considering a reference material, which is of same nature as the constituents of the heterogeneous material. We deduce that the estimates of the macroscopic strength domain are the solutions to self-consistent equations and we give their derivation procedure. To cite this article: S. Turgeman, B. Guessab, C. R. Mecanique 330 (2002) 623–626.     

Construction of positive definite cubature formulae and approximation of functions via Voronoi tessellations

Let Ω ⊂ ℝ ^d be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae of order two on Ω. Here we study extremal properties of those positive definite formulae that can be generated by a centroidal Voronoi tessellation of Ω. In this connection we come across a class of operators of the form L_n[f](x): = ?_i=1ⁿ f_i(x)(f(y_i) + á?f(y_i), x-y_i?)L_n[f](\boldsymbol{x}):= \sum_{i=1}^n \phi_i(\boldsymbol{x})(f(\boldsymbol{y}_i) + \langle\nabla f(\boldsymbol{y}_i), \boldsymbol{x}-\boldsymbol{y}_i\rangle), where y₁,..., y_n\boldsymbol{y}_1,\dots, \boldsymbol{y}_n are distinct points in Ω and {ϕ ₁, ..., ϕ _n } is a partition of unity on Ω. We present best possible pointwise error estimates and describe operators L _n with a smallest constant in an L ^p error estimate for 1 ≤ p < ∞ . For a generalization, we introduce a new type of Voronoi tessellation in terms of a twice continuously differentiable and strictly convex function f. It allows us to describe a best operator L _n for approximating f by L _n [f] with respect to the L ^p norm.     

Numerical cubature formulae with preassigned knots

Summary In this paper we give using the orthogonal polynomial theory, conditions ensuring the existence of cubature formulae with weight function on compact subsetsK in ² which have some given knots. The formulae are exact on the space of polynomials of two variablesx ₁ ,x ₂ with a degree not greater thanm _i +k _i ,i=1, 2.     

Cubature formulae which are exact on spacesP,intermediate betweenP k andQ k′

Summary In this paper we search, from the orthogonal polynomial theory, for conditions which allow to obtain cubature formulae on compacts of ⁿ , with weight function, and which are exact on the spaceR( _{k 1}, k₂, ..., k_n) of all polynomials of degree k _i respectively to each variablex _i , 1in.     

Sur les formules de quadrature numérique à nombre minimal de noeuds d'intégration

Résumè Cet article a pour objet la recherche, à partir de la théorie des polynômes orthogonaux, de conditions permettant l'obtention de formules de quadrature numérique sur des domaines de ⁿ, avec fonction poids, à nombre minimal de noeuds et exactes sur les espacesQ _k de polynômes de degré k par rapport à chacune de leurn variables. Ces résultats, complétés par des exemples numériques originaux dans ², adaptent à ces espacesQ _k ceux démontréq par H.J. Schmid [14] dans le cadre des espacesP _k de polynômes.

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About Cubature formulas with a minimal number of knots

Summary In this paper we search, from the orthogonal polynomial theory, for conditions which allow to obtain cubature formulas on sets of ⁿ, with weight function. which have a minimal number of knots and which are exact on the spaceQ _k of all polynomials of degree k with respect to each variablex _i, 1in.These results, completed by original numerical examples in ², adapt to the spacesQ _k those proved by H.J. Schmid [14] in the case of polynomial spacesP _k.

     

Characterization of the Existence of an Enriched Linear Finite Element Approximation Using Biorthogonal Systems

The present paper is intended to give a characterization of the existence of an enriched linear finite element approximation based on biorthogonal systems. It is shown that the enriched element exists if and only if a certain multivariate generalized trapezoidal type cubature formula has a nonzero approximation error. Furthermore, for such an enriched element we derive simple explicit formulas for the basis functions, and show that the approximation error can be written as the sum of the error of the (non-enriched) element plus a perturbation that depends on the enrichment function. Finally, we estimate the approximation error in L² norm. We also give an alternative approach to estimate the approximation error, which relies on an appropriate use of the Poincaré inequality.     

Convexity results and sharp error estimates in approximate multivariate integration

An interesting property of the midpoint rule and the trapezoidal rule, which is expressed by the so-called Hermite-Hadamard inequalities, is that they provide one-sided approximations to the integral of a convex function. We establish multivariate analogues of the Hermite-Hadamard inequalities and obtain access to multivariate integration formulae via convexity, in analogy to the univariate case. In particular, for simplices of arbitrary dimension, we present two families of integration formulae which both contain a multivariate analogue of the midpoint rule and the trapezoidal rule as boundary cases. The first family also includes a multivariate analogue of a Maclaurin formula and of the two-point Gaussian quadrature formula; the second family includes a multivariate analogue of a formula by P.C. Hammer and of Simpson's rule. In both families, we trace out those formulae which satisfy a Hermite-Hadamard inequality. As an immediate consequence of the latter, we obtain sharp error estimates for twice continuously differentiable functions.

     

A Definiteness Theory for Cubature Formulae of Order Two

Let Ω ⊂ ℝ^d be a compact convex set of positive measure. A cubature formula will be called positive definite (or a pd-formula, for short) if it approximates the integral ∫_Ω f(x) dx of every convex function f from below. The pd-formulae yield a simple sharp error bound for twice continuously differentiable functions. In the univariate case (d = 1), they are the quadrature formulae with a positive semidefinite Peano kernel of order two. As one of the main results, we show that there is a correspondence between pd-formulae and partitions of unity on Ω. This is a key for an investigation of pd-formulae without employing the complicated multivariate analogue of Peano kernels. After introducing a preorder, we establish criteria for maximal pd-formulae. We also find a lower bound for the error constant of an optimal pd-formula. Finally, we describe a phenomenon which resembles a property of Gaussian formulae.