Publications

2018

• A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems
  
  • Bourgeois Laurent
  • Recoquillay Arnaud
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2018, 52 (1), pp.123-145. This paper is dedicated to a new way of presenting the Tikhonov regularization in the form of a mixed formulation. Such formulation is well adapted to the regularization of linear ill-posed partial differential equations because when it comes to discretization, the mixed formulation enables us to use some standard finite elements. As an application of our theory, we consider an inverse obstacle problem in an acoustic waveguide. In order to solve it we use the so-called “exterior approach”, which couples the mixed formulation of Tikhonov regularization and a level set method. Some 2d numerical experiments show the feasibility of our approach. (10.1051/m2an/2018008)
  DOI : 10.1051/m2an/2018008
• A Family of Crouzeix-Raviart Finite Elements in 3D
  
  • Ciarlet Patrick
  • Dunkl Charles F
  • Sauter Stefan A
  Analysis and Applications, World Scientific Publishing, 2018. In this paper we will develop a family of non-conforming " Crouzeix-Raviart " type finite elements in three dimensions. They consist of local polynomials of maximal degree p ∈ N on simplicial finite element meshes while certain jump conditions are imposed across adjacent simplices. We will prove optimal a priori estimates for these finite elements. The characterization of this space via jump conditions is implicit and the derivation of a local basis requires some deeper theoretical tools from orthogonal polynomials on triangles and their representation. We will derive these tools for this purpose. These results allow us to give explicit representations of the local basis functions. Finally we will analyze the linear independence of these sets of functions and discuss the question whether they span the whole non-conforming space. (10.1142/S0219530518500070)
  DOI : 10.1142/S0219530518500070
• Accuracy of a Low Mach Number Model for Time-Harmonic Acoustics
  
  • Mercier Jean-François
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (4), pp.1891-1912. We study the time-harmonic acoustic radiation in a fluid in flow. To go beyond the convected Helmholtz equation adapted only to potential flows, starting from the Goldstein equations, coupling exactly the acoustic waves to the hydrodynamic field, we develop a new model in which the description of the hydrodynamic phenomena is simplified. This model, initially developed for a carrier flow of low Mach number M , is proved theoretically to be accurate, associated to a low error bounded by M 2 . Numerical experiments confirm the M 2 law and show that the model remains of very good quality for flow of moderate Mach numbers. (10.1137/17M113976X)
  DOI : 10.1137/17M113976X
• Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients
  
  • Ciarlet Patrick
  • Giret Léandre
  • Jamelot Erell
  • Kpadonou Félix D.
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2018, 52 (5), pp.2003-2035. We study first the convergence of the finite element approximation of the mixed diffusion equations with a source term, in the case where the solution is of low regularity. Such a situation commonly arises in the presence of three or more intersecting material components with different characteristics. Then we focus on the approximation of the associated eigenvalue problem. We prove spectral correctness for this problem in the mixed setting. These studies are carried out without, and then with a domain decomposition method. The domain decomposition method can be non-matching in the sense that the traces of the finite element spaces may not fit at the interface between subdomains. Finally, numerical experiments illustrate the accuracy of the method. (10.1051/m2an/2018011)
  DOI : 10.1051/m2an/2018011
• Asymptotic method for estimating magnetic moments from field measurements on a planar grid
  
  • Baratchart Laurent
  • Chevillard Sylvain
  • Leblond Juliette
  • Lima Eduardo Andrade
  • Ponomarev Dmitry
  , 2018. Scanning magnetic microscopes typically measure the vertical component B_3 of the magnetic field on a horizontal rectangular grid at close proximity to the sample. This feature makes them valuable instruments for analyzing magnetized materials at fine spatial scales, e.g., in geosciences to access ancient magnetic records that might be preserved in rocks by mapping the external magnetic field generated by the magnetization within a rock sample. Recovering basic characteristics of the magnetization (such as its net moment, i.e., the integral of the magnetization over the sample's volume) is an important problem, specifically when the field is too weak or the magnetization too complex to be reliably measured by standard bulk moment magnetometers. In this paper, we establish formulas, asymptotically exact when R goes large, linking the integral of x_1 B_3, x_2 B_3, and B_3 over a square region of size R to the first, second, and third component of the net moment (and higher moments), respectively, of the magnetization generating B_3. The considered square regions are centered at the origin and have sides either parallel to the axes or making a 45-degree angle with them. Differences between the exact integrals and their approximations by these asymptotic formulas are explicitly estimated, allowing one to establish rigorous bounds on the errors. We show how the formulas can be used for numerically estimating the net moment, so as to effectively use scanning magnetic microscopes as moment magnetometers. Illustrations of the method are provided using synthetic examples.