Implicit a posteriori error estimation in cut finite elements

Abstract

We describe a strategy for implicit a posteriori error estimation in cut finite elements. Our approach is based on the definition of local residual-driven corrector problems that use a local order elevation of the finite element space to construct a correction of the current approximation. The recovered higher-order accurate approximation is then used to construct error estimation in energy norm. We discuss implications of this scheme in the presence of cut elements, for instance regarding the construction of local corrector regions or the imposition of local boundary conditions. Combining the estimator with the finite cell method and a mesh refinement scheme, we numerically demonstrate its effectivity in terms of predicting the true error and its suitability to steer mesh adaptivity. Our results confirm that the estimation achieves the same effectivity in cut meshes as in standard boundary-fitted meshes, irrespective of the polynomial degree.