Comparing different versions of tiling cohomology

We establish direct isomorphisms between different versions of tiling cohomology. The first version is the direct limit of the cohomologies of the approximants in the Anderson-Putnam-Gähler system, the second is the recently introduced PV-cohomology of Savinien and Bellissard and the third is pattern equivariant cohomology. For the last two versions one can define weak cohomology groups. We show that the isomorphisms extend to the weak versions. This leads to an alternative formulation of the pattern equivariant mixed quotient group which describes deformations of the tiling modulo topological conjugacy.     

On the Stokes system and on the biharmonic equation in the half‐space: an approach via weighted Sobolev spaces

In this paper, we investigate the Stokes system and the biharmonic equation in a half‐space of ?ⁿ. Our approach rests on the use of a family of weighted Sobolev spaces as a framework for describing the behaviour at infinity. A complete class of existence, uniqueness and regularity results for both the problems is proved. The proofs are mainly based on the principle of reflection. Copyright © 2002 John Wiley & Sons, Ltd.     

The spectrum of a weighted Laplacian in the half‐space

J. Banasiak In this paper, we deal with spectral properties of a weighted Laplacian in the half‐space when a Dirichlet or a Neumann boundary condition is imposed. After proving that the spectrum is discrete under suitable assumptions, we give explicit formulae of eigenvalues and eigenfunctions in a specific case. In particular, the obtained eigenfunctions are rational or pseudo‐rational and have remarkable orthogonality properties. These results suggest the use of the discovered functions for approximating solutions of elliptic problems in the half‐space. Copyright © 2015 John Wiley & Sons, Ltd.     

Inverted finite elements for div-curl systems in the whole space

We use inverted finite element method (IFEM) for computing three-dimensional vector potentials and for solving div-curl systems in the whole space \(\mathbb {R}^{3}\). IFEM is substantially different from the existing approaches since it is a non truncature method which preserves the unboundness of the domain. After developping the method, we analyze its convergence in term of weighted norms. We then give some three-dimensional numerical results which demonstrate the efficiency and the accuracy of the method and confirm its convergence.     

Explicit div-curl inequalities in bounded and unbounded domains of $${{\mathbb {R}}}^3$$

We give a constructive proof of some functional inequalities related to the div and curl operators in bounded and unbounded domains of \({{\mathbb {R}}}^3\). Our new innovation consists in giving explicit constants in several geometric configurations. These inequalities are of a first use in solving div-curl systems and vector potential problems arising in physics.     

On the Laplace operator and on the vector potential problems in the half‐space: an approach using weighted spaces

The purpose of the present paper is twofold. The first object is to study the Laplace equation with inhomogeneous Dirichlet and Neumann boundary conditions in the half‐space of ?^N. The behaviour of solutions at infinity is described by means of a family of weighted Sobolev spaces. A class of existence, uniqueness and regularity results are obtained. The second purpose is to investigate some properties of grad, div and curl operators in order to treat curl–div systems of the form curl w = u, div w = 0 and problems related to vector potentials and Helmholtz decomposition.Copyright © 2003 John Wiley & Sons, Ltd.     

A mortar spectral element method for 3D Maxwell's equations

^** Email: Tahar.Boulmezaoud{at}univ-pau.fr^*** Email: Mohammed.Elrhabi{at}math.jussieu.fr In this paper we propose a mortar spectral element method forsolving Maxwell's equations in 3D bounded cavities. The methodis based on a non-conforming decomposition of the domain intothe union of non-overlapping parallelepipeds. After provingan error estimate, we present some 3D computational resultswhich confirm the performance of the method.