Convergence analysis of the $Q$1 -finite element method for elliptic problems with non-boundary-fitted meshes

1 LMTR - Laboratoire de Modélisation des interactions et Transferts en Réacteur
SMTA - Service Mesures et modélisation des Transferts et des Accidents graves : DEN/DTN
Abstract : The aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain’s boundary. This problematic of the last century (e.g. the finite difference methodology) returns to topical studies with the huge development of domain embedding, fictitious domain or Cartesiangrid methods. These methods use regular structured meshes (most often Cartesian) for non-aligned domains. Although non-boundary fitted approaches become more and more applied, very few studies are devoted to theoretical error estimates. In this paper, the convergence of a $Q$1-nonconforming finite element method is analyzed for secondorder elliptic problems with Dirichlet, Robin or Neumann boundary conditions. The finite element method uses standard $Q$1 rectangular finite elements. As the finite element approximate space is not contained in the original solution space, this method is referred to as nonconforming. A stairstep boundary defined from the Cartesian mesh approximates the original domain’s boundary. The convergence analysis of the finite element method for such a kind of non-boundary fitted stairstepped approximation is no treated in the literature. The study of Dirichlet problems is based on similar techniques as those classically used with boundary-fitted linear triangular finite elements. The estimates obtained for Robin problems are novel and use some more technical arguments. The rate of convergence is proved to be in $O$($h^{1/2}$) for the $H^1$ norm for all general boundary conditions, and classical duality arguments allow to obtain an $O(h)$ error estimate in the $L^2$ norm for Dirichlet problems. Numerical results obtained with fictitious domain techniques, that impose original boundary conditions on a non-boundary fitted approximate immersed interface, are presented. These results confirm the theoretical rates of convergence
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Isabelle Ramière. Convergence analysis of the $Q$1 -finite element method for elliptic problems with non-boundary-fitted meshes. International Journal for Numerical Methods in Engineering, Wiley, 2008, 75 (9), pp.1007-1052. ⟨10.1002/nme.2278⟩. ⟨cea-03581198⟩

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