Welcome to the TMR Project "Wavelets in Numerical Simulation"

RESEARCH TASKS

• Complex Geometries

  Adaptation of multiscale decompositions to more complex multidimensional geometries such as bounded domains and closed manifolds.
  Team in charge: Turin
  
  

• Non Coercive Problems

  Preconditioning for poorly elliptic, highly unisotropic or non-elliptic problems: this concerns convection-diffusion problems with strongly dominating convection or discretizations of saddle point problems based on compatible pairs of multiscale spaces, satisfying the Ladychenskaya-Babuska-Brezzi condition.
  Team in charge: Marseille
  
  

• Non linear terms

  Analysis and numerical approximation of nonlinear terms applied to multiscale expansions: so far only several ad-hoc methods for this central objective have been developed to deal with such nonlinear terms.
  Team in charge: Valencia
  
  

• Integral Equations

  Discretization of integral operator equations: the seminal work of Beylkin, Coifman and Rokhlin, has demonstrated the potential of wavelet bases for sparsifying dense matrices arising from the discretization of integral equations. However, a number of principal questions are still open which are of vital importance for realistic problems. This concerns preconditioning operators of order different from zero, constructing suitable wavelets on possibly non-smooth manifolds of general topology, their analysis and implementation.
  Team in charge: Zurich
  
  

• Coupling with other methods

  Coupling of wavelet-based methods with finite element methods. We plan to analyze the situations in which the two methods could interact in a constructive way. We expect also strong feedback and improved insight for classical discretization schemes.
  Team in charge: Lisbon
  
  

• Adaptivity

  Nonlinear approximation and adaptive solvers for PDEs and integral equations. Efficient and reliable adaptive schemes have to be developed also for poorly elliptic or non-elliptic problems as well as their time dependent variants. It is particularly important to incorporate appropriate problem-dependent error functionals that permit the accurate computation of quantities like drag and lift in fluid problems in a bottom-up-approach. This means to work up an approximate solution starting from coarse levels and tracking from the beginning only the significant parts of the wavelet expansion without ever going through the complexity of the full highest scale of resolution
  Team in charge: Aachen
  
  

• Applications

  Application of the methods developed to complex problems in fields as diverse as Fluid Dynamics, Elasticity, Chemical Engineering, Electromagnetism.
  Team in charge: Karlsruhe
  
  

• Project Software Database

  Team in charge: Paris