Hybrid coupling of CG and HDG discretizations based on Nitsche’s method

Abstract

A strategy to couple continuous Galerkin (CG) and hybridizable discontinuous Galerkin (HDG) discretizations based only on the HDG hybrid variable is presented for linear thermal and elastic problems. The hybrid CG–HDG coupling exploits the definition of the numerical flux and the trace of the solution on the mesh faces to impose the transmission conditions between the CG and HDG subdomains. The continuity of the solution is imposed in the CG problem via Nitsche’s method, whereas the equilibrium of the flux at the interface is naturally enforced as a Neumann condition in the HDG global problem. The proposed strategy does not affect the core structure of CG and HDG discretizations. In fact, the resulting formulation leads to a minimally-intrusive coupling, suitable to be integrated in existing CG and HDG libraries. Numerical experiments in two and three dimensions show optimal global convergence of the stress and superconvergence of the displacement field, locking-free approximation, as well as the potential to treat structural problems of engineering interest featuring multiple materials with compressible and nearly incompressible behaviors.

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Acknowledgements

This work is partially supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie actions (Grant Nos. 675919 and 764636), the Spanish Ministry of Economy and Competitiveness (Grant No. DPI2017-85139-C2-2-R) and the Generalitat de Catalunya (Grant No. 2017-SGR-1278). Andrea La Spina is supported by the European Education, Audiovisual and Culture Executive Agency (EACEA) under the Erasmus Mundus Joint Doctorate Simulation in Engineering and Entrepreneurship Development (SEED), FPA 2013-0043.

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La Spina, A., Giacomini, M. & Huerta, A. Hybrid coupling of CG and HDG discretizations based on Nitsche’s method. Comput Mech 65, 311–330 (2020). https://doi.org/10.1007/s00466-019-01770-8

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Keywords

  • Hybridizable discontinuous Galerkin
  • Coupling with finite element
  • Nitsche’s method
  • Locking-free
  • Superconvergence