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Stability Analysis of Polytopic Discontinuous Galerkin Approximations of the Stokes Problem with Applications to Fluid–Structure Interaction Problems

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Abstract

We present a stability analysis of the Discontinuous Galerkin method on polygonal and polyhedral meshes (PolyDG) for the Stokes problem. In particular, we analyze the discrete inf-sup condition for different choices of the polynomial approximation order of the velocity and pressure approximation spaces. To this aim, we employ a generalized inf-sup condition with a pressure stabilization term. We also prove a priori hp-version error estimates in suitable norms. We numerically check the behaviour of the inf-sup constant and the order of convergence with respect to the mesh configuration, the mesh-size, and the polynomial degree. Finally, as a relevant application of our analysis, we consider the PolyDG approximation for a 2D fluid–structure interaction problem and we numerically explore the stability properties of the method.

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Authors and Affiliations

  1. MOX, Dipartimento di Matematica, Politecnico di Milano, Milan, Italy

    Paola F. Antonietti, Marco Verani & Stefano Zonca

  2. Fakultät für Mathematik, Universität Wien, Wien, Austria

    Lorenzo Mascotto

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  1. Paola F. Antonietti
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  2. Lorenzo Mascotto
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  3. Marco Verani
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  4. Stefano Zonca
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Correspondence to Stefano Zonca.

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PFA, LM, MV and SZ are member of the INdAM Research group GNCS and this work is partially funded by INDAM-GNCS. PFA, MV and SZ have been partially funded by the PRIN Italian research grant n. 201744KLJL funded by MIUR. LM acknowledges the support of the Austrian Science Fund (FWF) project P33477.

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Antonietti, P.F., Mascotto, L., Verani, M. et al. Stability Analysis of Polytopic Discontinuous Galerkin Approximations of the Stokes Problem with Applications to Fluid–Structure Interaction Problems. J Sci Comput 90, 23 (2022). https://doi.org/10.1007/s10915-021-01695-6

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