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Implicit a posteriori error estimation in cut finite elements

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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.

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Acknowledgements

Haohan Sun gratefully acknowledges support from Tsinghua University Tutor Research Fund that funded his short-term stay at the University of Minnesota. Dominik Schillinger acknowledges support from the National Science Foundation via the NSF CAREER Award No. 1651577, from the German Research Foundation through the DFG Emmy Noether Award SCH 1249/2-1, and from the European Research Council via the ERC Starting Grant “ImageToSim” (Action No. 759001).

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  1. Department of Civil Engineering, Tsinghua University, 100084, Beijing, China

    Haohan Sun & Si Yuan

  2. Department of Civil Engineering and Geodetic Science, Institute of Mechanics and Computational Mechanics, Leibniz University Hannover, 30167, Hannover, Germany

    Dominik Schillinger

  3. Department of Civil, Environmental, and Geo-Engineering, University of Minnesota, 500 Pillsbury Drive S.E., Minneapolis, MN, 55455-0116, USA

    Dominik Schillinger

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Sun, H., Schillinger, D. & Yuan, S. Implicit a posteriori error estimation in cut finite elements. Comput Mech 65, 967–988 (2020). https://doi.org/10.1007/s00466-019-01803-2

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