CutFEM without cutting the mesh cells: A new way to impose Dirichlet and Neumann boundary conditions on unfitted meshes

Abstract We present a method of CutFEM type for the Poisson problem with either Dirichlet or Neumann boundary conditions. The computational mesh is obtained from a background (typically uniform Cartesian) mesh by retaining only the elements intersecting the domain where the problem is posed. The resulting mesh does not thus fit the boundary of the problem domain. Several finite element methods (XFEM, CutFEM) adapted to such meshes have been recently proposed. The originality of the present article consists in avoiding integration over the elements cut by the boundary of the problem domain, while preserving the optimal convergence rates, as confirmed by both the theoretical estimates and the numerical results.

[1]  R. Glowinski,et al.  A fictitious domain method for Dirichlet problem and applications , 1994 .

[2]  R. Glowinski,et al.  Error analysis of a fictitious domain method applied to a Dirichlet problem , 1995 .

[3]  Peter Hansbo,et al.  Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem , 2014 .

[4]  Roland Glowinski,et al.  Error estimates for fictitious domain/penalty/finite element methods , 1992 .

[5]  T. Hughes,et al.  Two classes of mixed finite element methods , 1988 .

[6]  R. Glowinski,et al.  A distributed Lagrange multiplier/fictitious domain method for particulate flows , 1999 .

[7]  Frédéric Hecht,et al.  New development in freefem++ , 2012, J. Num. Math..

[8]  T. Belytschko,et al.  MODELING HOLES AND INCLUSIONS BY LEVEL SETS IN THE EXTENDED FINITE-ELEMENT METHOD , 2001 .

[9]  R. Glowinski,et al.  Distributed Lagrange multipliers based on fictitious domain method for second order elliptic problems , 2007 .

[10]  B. Maury,et al.  A smooth extension method , 2013 .

[11]  Bertrand Maury.  Numerical Analysis of a Finite Element/Volume Penalty Method , 2009, SIAM J. Numer. Anal..

[12]  Peter Hansbo,et al.  A cut finite element method with boundary value correction , 2015, Math. Comput..

[13]  P. Hansbo,et al.  Fictitious domain finite element methods using cut elements , 2012 .

[14]  Jaroslav Haslinger,et al.  A New Fictitious Domain Approach Inspired by the Extended Finite Element Method , 2009, SIAM J. Numer. Anal..

[15]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[16]  Jean-François Remacle,et al.  Imposing Dirichlet boundary conditions in the eXtended Finite Element Method , 2011 .

[17]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[18]  I. Babuska,et al.  A DiscontinuoushpFinite Element Method for Diffusion Problems , 1998 .

[19]  Rolf Stenberg,et al.  Nitsche's method for general boundary conditions , 2009, Math. Comput..

[20]  Peter Hansbo,et al.  Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method , 2010 .

[21]  G. Burton.  Sobolev Spaces , 2013 .

[22]  Peter Hansbo,et al.  CutFEM: Discretizing geometry and partial differential equations , 2015 .