A new ϕ -FEM approach for problems with natural boundary conditions
Duprez, Michel; Lleras, Vanessa; Lozinski, Alexei (2020), A new ϕ -FEM approach for problems with natural boundary conditions. https://basepub.dauphine.psl.eu/handle/123456789/22338
TypeDocument de travail / Working paper
External document linkhttps://hal.archives-ouvertes.fr/hal-02521042
Series titleCahier de recherche du CEREMADE
MetadataShow full item record
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Computational Anatomy and Simulation for Medicine [MIMESIS]
Institut Montpelliérain Alexander Grothendieck [IMAG]
Laboratoire de Mathématiques de Besançon (UMR 6623) [LMB]
Abstract (EN)We present a new finite element method, called φ-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, φ-FEM does not need any non-standard 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 φ-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 / KeywordsFinite element method; fictitious domain; level-set; Neumann conditions; immersed boundary method
Showing items related by title and author.
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
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
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