• xmlui.mirage2.page-structure.header.title
    • français
    • English
  • Help
  • Login
  • Language 
    • Français
    • English
View Item 
  •   BIRD Home
  • CEREMADE (UMR CNRS 7534)
  • CEREMADE : Publications
  • View Item
  •   BIRD Home
  • CEREMADE (UMR CNRS 7534)
  • CEREMADE : Publications
  • View Item
JavaScript is disabled for your browser. Some features of this site may not work without it.

Browse

BIRDResearch centres & CollectionsBy Issue DateAuthorsTitlesTypeThis CollectionBy Issue DateAuthorsTitlesType

My Account

LoginRegister

Statistics

Most Popular ItemsStatistics by CountryMost Popular Authors
Thumbnail

A new ϕ-FEM approach for problems with natural boundary conditions

Duprez, Michel; Lleras, Vanessa; Lozinski, Alexei (2023), A new ϕ-FEM approach for problems with natural boundary conditions, Numerical Methods for Partial Differential Equations, 39, 1, p. 281-303. 10.1002/num.22878

View/Open
duprez.pdf (644.9Kb)
Type
Article accepté pour publication ou publié
Date
2023
Journal name
Numerical Methods for Partial Differential Equations
Volume
39
Number
1
Published in
Paris
Pages
281-303
Publication identifier
10.1002/num.22878
Metadata
Show full item record
Author(s)
Duprez, Michel
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Lleras, Vanessa
Institut Montpelliérain Alexander Grothendieck [IMAG]
Lozinski, Alexei
Laboratoire de Mathématiques de Besançon (UMR 6623) [LMB]
Abstract (EN)
We present a new finite element method, called -FEM, to solve numerically elliptic partial differential equations with natural (Neumann or Robin) boundary conditions using simple computational grids, not fitted to the boundary of the physical domain. The boundary data are taken into account using a level-set function, which is a popular tool to deal with complicated or evolving domains. Our approach belongs to the family of fictitious domain methods (or immersed boundary methods) and is close to recent methods of CutFEM/XFEM type. Contrary to the latter, -FEM does not need any nonstandard numerical integration on cut mesh elements or on the actual boundary, while assuring the optimal convergence orders with finite elements of any degree and providing reasonably well conditioned discrete problems. In the first version of -FEM, only essential (Dirichlet) boundary conditions was considered. Here, to deal with natural boundary conditions, we introduce the gradient of the primary solution as an auxiliary variable. This is done only on the mesh cells cut by the boundary, so that the size of the numerical system is only slightly increased. We prove theoretically the optimal convergence of our scheme and a bound on the discrete problem conditioning, independent of the mesh cuts. The numerical experiments confirm these results.
Subjects / Keywords
Finite element method; fictitious domain; level-set; Neumann conditions

Related items

Showing items related by title and author.

  • Thumbnail
    A new $\phi$-FEM approach for problems with natural boundary conditions 
    Duprez, Michel; Lleras, Vanessa; Lozinski, Alexei; Abbad, Narima (2022) Article accepté pour publication ou publié
  • Thumbnail
    φ-FEM: a finite element method on domains defined by level-sets 
    Duprez, Michel; Lozinski, Alexei (2020) Article accepté pour publication ou publié
  • Thumbnail
    A new approach for crew pairing problems by column generation with an application to air transportation 
    Lavoie, Sylvie; Minoux, Michel; Odier, Edouard (1988) Article accepté pour publication ou publié
  • Thumbnail
    Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions 
    Da Lio, Francesca; Lions, Pierre-Louis; Barles, Guy; Souganidis, Panagiotis E. (2008) Article accepté pour publication ou publié
  • Thumbnail
    Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints 
    Lasry, Jean-Michel; Lions, Pierre-Louis (1989) Article accepté pour publication ou publié
Dauphine PSL Bibliothèque logo
Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16
Phone: 01 44 05 40 94
Contact
Dauphine PSL logoEQUIS logoCreative Commons logo