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A Superconvergent HDG Method for Stokes Flow with Strongly Enforced Symmetry of the Stress Tensor

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Abstract

This work proposes a superconvergent hybridizable discontinuous Galerkin (HDG) method for the approximation of the Cauchy formulation of the Stokes equation using same degree of polynomials for the primal and mixed variables. The novel formulation relies on the well-known Voigt notation to strongly enforce the symmetry of the stress tensor. The proposed strategy introduces several advantages with respect to the existing HDG formulations. First, it remedies the suboptimal behavior experienced by the classical HDG method for formulations involving the symmetric part of the gradient of the primal variable. The optimal convergence of the mixed variable is retrieved and an element-by-element postprocess procedure leads to a superconvergent velocity field, even for low-order approximations. Second, no additional enrichment of the discrete spaces is required and a gain in computational efficiency follows from reducing the quantity of stored information and the size of the local problems. Eventually, the novel formulation naturally imposes physical tractions on the Neumann boundary. Numerical validation of the optimality of the method and its superconvergent properties is performed in 2D and 3D using meshes of different element types.

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Authors and Affiliations

  1. Laboratori de Càlcul Numèric (LaCàN), ETS de Ingenieros de Caminos, Canales y Puertos, Universitat Politècnica de Catalunya, Barcelona, Spain

    Matteo Giacomini, Alexandros Karkoulias & Antonio Huerta

  2. Centre Internacional de Metodes Numerics en Enyinyeria (CIMNE), Campus Nord UPC, Barcelona, Spain

    Alexandros Karkoulias

  3. Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Wales, UK

    Ruben Sevilla

Authors
  1. Matteo Giacomini
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  2. Alexandros Karkoulias
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  3. Ruben Sevilla
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  4. Antonio Huerta
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Corresponding author

Correspondence to Matteo Giacomini.

Additional information

This work was partially supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 675919 and the Spanish Ministry of Economy and Competitiveness (Grant number: DPI2017-85139-C2-2-R). The support of the Generalitat de Catalunya (Grant number: 2017SGR1278) is also gratefully acknowledged. Finally, Alexandros Karkoulias was 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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Giacomini, M., Karkoulias, A., Sevilla, R. et al. A Superconvergent HDG Method for Stokes Flow with Strongly Enforced Symmetry of the Stress Tensor. J Sci Comput 77, 1679–1702 (2018). https://doi.org/10.1007/s10915-018-0855-y

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