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Robust Interior Penalty Discontinuous Galerkin Methods

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Abstract

A new variant of the IPDG method is presented, involving carefully constructed weighted averages of the gradient of the approximate solution. The method is shown to be robust even for the most extreme simultaneous local mesh, polynomial degree and diffusion coefficient variation scenarios, without resulting into unreasonably large penalization. The new IPDG method, henceforth termed as robust IPDG (RIPDG), offers typically significantly better error behaviour and conditioning than the standard IPDG method when applied to scenarios with strong mesh/polynomial degree/diffusion local variation, especially when the underlying approximation space does not contain a sufficiently rich conforming subspace. The latter is of particular importance in the context of IPDG methods on polygonal/polyhedral meshes. On the other hand, when using uniform meshes, constant polynomial degree for problems with constant diffusion coefficients the RIPDG method is identical to the classical IPDG. Numerical experiments indicate the favourable performance of the new RIPDG method over the classical version in terms of conditioning and error.

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References

  1. Annavarapu, C., Hautefeuille, M., Dolbow, J.E.: A robust Nitsche’s formulation for interface problems. Comput. Methods Appl. Mech. Engrg. 225(228), 44–54 (2012)

    Article  MathSciNet  Google Scholar 

  2. Arnold, D.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)

    Article  MathSciNet  Google Scholar 

  3. Babuška, I.: The finite element method with penalty. Math. Comp. 27, 221–228 (1973)

    Article  MathSciNet  Google Scholar 

  4. Baker, G.: Finite element methods for elliptic equations using nonconforming elements. Math. Comp. 31, 45–59 (1977)

    Article  MathSciNet  Google Scholar 

  5. Bassi, F., Botti, L., Colombo, A., Di Pietro, D., Tesini, P.: On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations. J. Comput. Phys. 231, 45–65 (2012)

    Article  MathSciNet  Google Scholar 

  6. Burman, E., Zunino, P.: A domain decomposition method based on weighted interior penalties for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 44, 1612–1638 (2006)

    Article  MathSciNet  Google Scholar 

  7. Cangiani, A., Dong, Z., Georgoulis, E., Houston, P.: \(hp\)–Version discontinuous Galerkin methods for advection–diffusion–reaction problems on polytopic meshes. ESAIM: M2AN 50, 699–725 (2016)

    Article  MathSciNet  Google Scholar 

  8. Cangiani, A., Dong, Z., Georgoulis, E.H.: \(hp\)-version space-time discontinuous Galerkin methods for parabolic problems on prismatic meshes. SIAM J. Sci. Comput. 39, A1251–A1279 (2017)

    Article  MathSciNet  Google Scholar 

  9. Cangiani, A., Dong, Z., Georgoulis, E.H.: \(hp\)-version discontinuous Galerkin methods on essentially arbitrarily-shaped elements. Math. Comp. 91, 1–35 (2022)

    Article  MathSciNet  Google Scholar 

  10. Cangiani, A., Dong, Z., Georgoulis, E.H., Houston, P.: \(hp\)-version discontinuous Galerkin methods on polygonal and polyhedral meshes. SpringerBriefs in Mathematics, Springer, Cham (2017)

    Book  Google Scholar 

  11. Cangiani, A., Georgoulis, E., Houston, P.: \(hp\)-Version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 24, 2009–2041 (2014)

    Article  MathSciNet  Google Scholar 

  12. Chernov, A.: Optimal convergence estimates for the trace of the polynomial \(L^{2}\)-projection operator on a simplex. Math. Comp. 81, 765–787 (2012)

    Article  MathSciNet  Google Scholar 

  13. Dong, Z.: Discontinuous galerkin methods for the biharmonic problem on polygonal and polyhedral meshes. Int. J. Numer. Anal. Model. 16, 825–846 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Dong, Z.: On the exponent of exponential convergence of \(p\)-version FEM spaces. Adv. Comput. Math. 45, 757–785 (2019)

    Article  MathSciNet  Google Scholar 

  15. Dryja, M.: On discontinuous Galerkin methods for elliptic problems with discontinuous coefficients, vol. 3, pp. 76–85 (2003). Dedicated to Raytcho Lazarov

  16. Ern, A., Stephansen, A.F., Zunino, P.: A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity. IMA J. Numer. Anal. 29, 235–256 (2009)

    Article  MathSciNet  Google Scholar 

  17. Georgoulis, E.: Discontinuous Galerkin methods on shape-regular and anisotropic meshes, D.Phil. Thesis, University of Oxford (2003)

  18. Georgoulis, E., Lasis, A.: A note on the design of \(hp\)-version interior penalty discontinuous Galerkin finite element methods for degenerate problems. IMA J. Numer. Anal. 26, 381–390 (2006)

    Article  MathSciNet  Google Scholar 

  19. Gudi, T.: A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comp. 79, 2169–2189 (2010)

    Article  MathSciNet  Google Scholar 

  20. Gustafsson, T., Stenberg, R., Videman, J.: On Nitsche’s method for elastic contact problems. SIAM J. Sci. Comput. 42, B425–B446 (2020)

    Article  MathSciNet  Google Scholar 

  21. Houston, P., Schwab, C., Süli, E.: Discontinuous \(hp\)-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39, 2133–2163 (2002)

    Article  MathSciNet  Google Scholar 

  22. Melenk, J., Schwab, C.: An \(hp\)-finite element method for convection-diffusion problems in one dimension. IMA J. Numer. Anal. 19, 425–453 (1999)

    Article  MathSciNet  Google Scholar 

  23. Nitsche, J.: Über ein variationsprinzip zur lösung von dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind. In: Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, vol. 36, pp. 9–15 1971 (Springer)

  24. Oleinik, O., Radkevič, E.: Second Order Equations with Nonnegative Characteristic Form. American Mathematical Society Providence (1973)

  25. Rivière, B., Wheeler, M., Girault, V.: A priori error estimates for finite element methods based on discontinuous approximation spaces for elliptic problems. SIAM J. Numer. Anal. 39, 902–931 (2001)

    Article  MathSciNet  Google Scholar 

  26. Veeser, A., Zanotti, P.: Quasi-optimal nonconforming methods for symmetric elliptic problems. III—Discontinuous Galerkin and other interior penalty methods. SIAM J. Numer. Anal. 56, 2871–2894 (2018)

    Article  MathSciNet  Google Scholar 

  27. Warburton, T., Hesthaven, J.S.: On the constants in \(hp\)-finite element trace inverse inequalities. Comput. Methods Appl. Mech. Engrg. 192, 2765–2773 (2003)

    Article  MathSciNet  Google Scholar 

  28. Wheeler, M.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15, 152–161 (1978)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Inria, 2 rue Simone Iff, 75589, Paris, France

    Zhaonan Dong

  2. CERMICS, Ecole des Ponts, 77455, Marne-la-Vallée, France

    Zhaonan Dong

  3. School of Computing and Mathematical Sciences, University of Leicester, University Road, Leicester, LE1 7RH, UK

    Emmanuil H. Georgoulis

  4. Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou, 15780, Greece

    Emmanuil H. Georgoulis

  5. IACM-FORTH, Crete, Greece

    Emmanuil H. Georgoulis

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  1. Zhaonan Dong
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  2. Emmanuil H. Georgoulis
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Correspondence to Zhaonan Dong.

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Dong, Z., Georgoulis, E.H. Robust Interior Penalty Discontinuous Galerkin Methods. J Sci Comput 92, 57 (2022). https://doi.org/10.1007/s10915-022-01916-6

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  • DOI: https://doi.org/10.1007/s10915-022-01916-6

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