A posteriori error estimates with boundary correction for a cut finite element method

Abstract

In this work we introduce, analyze and implement a residual-based a posteriori error estimation for the CutFEM fictitious domain method applied to an elliptic model problem. We consider the problem with smooth (nonpolygonal) boundary and, therefore, the analysis takes into account both the geometry approximation error on the boundary and the numerical approximation error. Theoretically, we can prove that the error estimation is both reliable and efficient. Moreover, the error estimation is robust in the sense that both the reliability and efficiency constants are independent of the arbitrary boundary-mesh intersection.

Flux recovery for Cut Finite Element Method and its application in a posteriori error estimation

“…The computation of this term, however, is not trivial and requires the construction of a sub mesh. Nevertheless, numerical results have shown that such error is not necessary to compute since the boundary approximation error can already be captured by the residual based error estimator in [18]. In this paper, we also discard such term and the numerical results also confirm that our error estimator is able to catch both the boundary approximation errors and the discretization error due to the numerical method.…”

“…Remark 4.4. From the above estimate, we observe that can be bounded by the residual based error estimator given in [18]. Moreover, the constants are uniformly bounded and independent of the domain-mesh intersection.…”

“…This assumption is necessary for the constant in the a posteriori error estimates to be independent of the geometry/mesh configuration see [18]. It is however not enough to guarantee the optimal a priori error estimates, which requires [16].…”

“…It is well known that AMR is extremely useful for problems with singularities, discontinuities, sharp derivatives etc.. And it has been extensively studied in the last several decades, see e.g., [41,3]. In [18], a residual based a posterior error estimator for the CutFEM method was studied. One drawback of residual based error estimation is that its reliability constants are unknown and usually not polynomial-robust and problem dependent.…”

“…In this work, we approximate the physical geometry by a piecewise affine polygonal domain. In [18], a boundary correction error was separated that particularly estimates the geometry approximation error. The computation of this term, however, is not trivial and requires the construction of a sub mesh.…”