A new $\phi$-FEM approach for problems with natural boundary conditions
Résumé
We present a new finite element method, called φ-FEM, to solve numerically elliptic partial differ-ential equations with natural (Neumann or Robin) boundary conditions using simple computationalgrids, not fitted to the boundary of the physical domain. The boundary data are taken into accountusing a level-set function, which is a popular tool to deal with complicated or evolving domains. Ourapproach belongs to the family of fictitious domain methods (or immersed boundary methods) and isclose to recent methods of cutFEM/XFEM type. Contrary to the latter, φ-FEM does not need anynon-standard numerical integration on cut mesh elements or on the actual boundary, while assuringthe optimal convergence orders with finite elements of any degree and providing reasonably well condi-tioned discrete problems. In the first version of φ-FEM, only essential (Dirichlet) boundary conditionswas considered. Here, to deal with natural boundary conditions, we introduce the gradient of theprimary solution as an auxiliary variable. This is done only on the mesh cells cut by the boundary, sothat the size of the numerical system is only slightly increased . We prove theoretically the optimalconvergence of our scheme and a bound on the discrete problem conditioning, independent of the meshcuts. The numerical experiments confirm these results.
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