Abstract
In this article, we aim to recover locally conservative and H(div) conforming fluxes for the linear Cut Finite Element Solution with Nitsche’s method for Poisson problems with Dirichlet boundary condition. The computation of the conservative flux in the Raviart–Thomas space is completely local and does not require to solve any mixed problem. The L2-norm of the difference between the numerical flux and the recovered flux can then be used as a posteriori error estimator in the adaptive mesh refinement procedure. Theoretically we also prove the global reliability and local efficiency. The theoretical results are verified in the numerical results. Moreover, in the numerical results we also observe optimal convergence rate for the flux error.
Highlights
Cut finite element method (CutFEM) may be regarded as a fictitious domain method
Thanks to the sharp reliability, equilibrate flux recovery has been extensively studied for various finite element methods on fitted meshes in the last decade
In order to achieve the same accuracy as the classical fitted method, boundary geometry and boundary data for CutFEM need to be approximated to similar accuracy
Summary
Cut finite element method (CutFEM) may be regarded as a fictitious domain method. The finite element fictitious domain method was introduced in [28] as an approach to simplify the meshing problem and further improved in [5, 8, 14, 30, 31]. Thanks to the sharp reliability, equilibrate flux recovery has been extensively studied for various finite element methods on fitted meshes in the last decade. Regarding the conservative numerical flux, its local construction is similar for the interior elements not cut by the boundary, it needs extra treatment for the ghost penalty term. In order to achieve the same accuracy as the classical fitted method, boundary geometry and boundary data for CutFEM need to be approximated to similar accuracy It is important in this context to derive error estimators that are able to integrate both the discretization error of the method and the discretization error of the geometry.
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