A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension

^** Email: eymard{at}math.univ-mlv.fr^*** Email: gallouet{at}cmi.univ-mrs.fr^**** Corresponding author. Email: herbin{at}cmi.univ-mrs.fr Finite-volume methods for problems involving second-order operatorswith full diffusion matrix can be used thanks to the definitionof a discrete gradient for piecewise constant functions on unstructuredmeshes satisfying an orthogonality condition. This discretegradient is shown to satisfy a strong convergence property forthe interpolation of regular functions, and a weak one for functionsbounded in a discrete H¹-norm. To highlight the importance ofboth properties, the convergence of the finite-volume schemefor a homogeneous Dirichlet problem with full diffusion matrixis proven, and an error estimate is provided. Numerical testsshow the actual accuracy of the method.     

Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions

By taking as a “prototype problem” a one-delay linear autonomous system of delay differential equations we present the problem of computing the characteristic roots of a retarded functional differential equation as an eigenvalue problem for a derivative operator with non-local boundary conditions given by the particular system considered. This theory can be enlarged to more general classes of functional equations such as neutral delay equations, age-structured population models and mixed-type functional differential equations.It is thus relevant to have a numerical technique to approximate the eigenvalues of derivative operators under non-local boundary conditions. In this paper we propose to discretize such operators by pseudospectral techniques and turn the original eigenvalue problem into a matrix eigenvalue problem. This approach is shown to be particularly efficient due to the well-known “spectral accuracy” convergence of pseudospectral methods. Numerical examples are given.     

Type I operators and the approximation of singular two-point boundary value problems

A class of self adjoint operators associated with second order singular ordinary differential expressions, arises naturally when the problem is weakly formulated and integration by parts is performed. We call this class Type I operators. It turns out that this class can be successfully used to tackle numerical approximations of singular two-point boundary value problems. They can also be approximated by regular differential operators in a straightforward manner without having to bring the delicate structure of singular differential operators to the forefront of the investigation.     

The equivalence relations between the best approximation and linear operators approximation in a Besov space

In this paper, we characterize Besov type space introduced by the best approximation, best approximating elements and a kind of linear operators. The project is supported by NSFC     

WEIGHTED APPROXIMATION ON SZASZ—TYPE OPERATORS

In this paper, we use weighted modules ω(?)(f,t)w to study the pointwise approximation on Szasz-type operators, and obtain the direct and converse theorem, as well as characterizations of the pointwise approximation of Jacobi-weighted Szasz-type operators.     

The new approximation operators with sigmoidal functions

The aim of this paper is to investigate approximation operators with logarithmic sigmoidal function of a class of two neural networks weights and a class of quasi-interpolation operators. Using these operators as approximation tools, the upper bounds of estimate errors are estimated for approximating continuous functions.     

The boundary element spline collocation for nonuniform meshes on the torus

The spline collocation method for a class of biperiodic strongly elliptic pseudodifferential operators is considered. As trial functions tensor products of odd degree splines are used and the collocation is imposed at the nodal points of the tensor product mesh. It is shown that the collocation problem is uniquely solvable if the maximum mesh length is small enough. Moreover, the approximation is stable and quasioptimal with respect to a norm depending on the order of the operator and the degree of approximating splines. Some convergence results are given for general and quasiuniform meshes. The results cover for example the single layer and the hypersingular operators.     

The Poisson boundary of covering Markov operators

Covering Markov operators are a measure theoretical generalization of both random walks on groups and the Brownian motion on covering manifolds. In this general setup we obtain several results on ergodic properties of their Poisson boundaries, in particular, that the Poisson boundary is always infinite if the deck group is non-amenable, and that the deck group action on the Poisson boundary is amenable. For corecurrent operators we show that the Radon-Nikodym cocycles of two quotients of the Poisson boundary are cohomologous iff these quotients coincide. It implies that the Poisson boundary is either purely non-atomic or trivial, and that the action of any normal subgroup of the deck group on the Poisson boundary is conservative. We show that the Poisson boundary is trivial for any corecurrent covering operator with a nilpotent (or, more generally, hypercentral) deck group. Other applications and examples are discussed. Supported by a British SERC Advanced Fellowship. A part of this work was done during my stay at MSRI, Berkeley supported by NSF Grant DMS 8505550.     

Harmonic approximation of difference operators

For a general class of difference operators Hε=Tε+Vε on ?²(d(εZ)), where Vε is a multi-well potential and ε is a small parameter, we analyze the asymptotic behavior as ε→0 of the (low-lying) eigenvalues and eigenfunctions. We show that the first n eigenvalues of Hε converge to the first n eigenvalues of the direct sum of harmonic oscillators on Rd located at the several wells. Our proof is microlocal.     

Discrete approximation of integral operators

A method to approximate the eigenvalues of linear operators depending on an unknown distribution is introduced and applied to weighted sums of squared normally distributed random variables. This area of application includes the approximation of the asymptotic null distribution of certain degenerated U- and V-statistics.

     

The weak metric approximation property and ideals of operators

We study the weak metric approximation property introduced by Lima and Oja. We show that a Banach space X has the weak metric approximation property if and only if F(Y,X), the space of finite rank operators, is an ideal in W(Y,X∗∗), the space of weakly compact operators for all Banach spaces Y.     

The construction and approximation of some neural networks operators

In this paper,the technique of approximate partition of unity is used to construct a class of neural networks operators with sigmoidal functions.Using the modulus of continuity of function as a metric,...     

Mixed hp-FEM on anisotropic meshes II: Hanging nodes and tensor products of boundary layer meshes

Summary. The divergence stability of mixed hp Finite Element Methods for incompressible fluid flow is analyzed. A discrete inf-sup condition is proved for a general class of meshes. The meshes may be refined anisotropically, geometrically and may contain hanging nodes on geometric patches. The inf-sup constant is shown to be independent of the aspect ratio of the anisotropic elements and the dependence on the polynomial degree is analyzed. Numerical estimates of inf-sup constants confirm the theoretical results. Received October 13, 1997 / Revised version received June 8, 1998 / Published online July 28, 1999     

Statistical approximation properties of q-Baskakov-Kantorovich operators

In the present paper we introduce a q-analogue of the Baskakov-Kantorovich operators and investigate their weighted statistical approximation properties. By using a weighted modulus of smoothness, we give some direct estimations for error in case 0 < q < 1.