Our site uses Javascript to enhance its usability.You can disable your ad blocker or whitelist our website www.worldscientific.com to view the full content.

Select your blocker:

Adblock Plus Instructions

  1. Click the AdBlock Plus icon in the extension bar
  2. Click the blue power button
  3. Click refresh
World Scientific
  • Search
    Quick Search in Journals
    Access type:
    Quick Search anywhere
    Access type:
    Advanced Search
  • My Cart
  •   
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More
×

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at [email protected] for any enquiries.
Open Access

A discrete three-dimensional divdiv complex on polyhedral meshes with application to a mixed formulation of the biharmonic problem

    https://doi.org/10.1142/S0218202524500313Cited by:0 (Source: Crossref)
    Next

    Abstract

    In this work, following the Discrete de Rham (DDR) paradigm, we develop an arbitrary-order discrete divdiv complex on general polyhedral meshes. The construction rests on (1) discrete spaces that are spanned by vectors of polynomials whose components are attached to mesh entities and (2) discrete operators obtained mimicking integration by parts formulas. We provide an in-depth study of the algebraic properties of the local complex, showing that it is exact on mesh elements with trivial topology. The new DDR complex is used to design a numerical scheme for the approximation of biharmonic problems, for which we provide detailed stability and convergence analyses. Numerical experiments complete the theoretical results.

    Communicated by L. Beirao da Veiga

    AMSC: 65N30, 65N99, 31B30

    References

    • 1. S. Adams and B. Cockburn , A mixed finite element method for elasticity in three dimensions, J. Sci. Comput. 25 (2005) 515–521. Crossref, Web of Science, Google Scholar
    • 2. P. F. Antonietti, A. Cangiani, J. Collis, Z. Dong, E. H. Georgoulis, S. Giani and P. Houston , Review of discontinuous Galerkin finite element methods for partial differential equations on complicated domains, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Lecture Notes in Computational Science and Engineering, Vol. 114 (Springer, 2016), pp. 279–308. Crossref, Google Scholar
    • 3. P. F. Antonietti, S. Giani and P. Houston , hp-version composite discontinuous Galerkin methods for elliptic problems on complicated domains, SIAM J. Sci. Comput. 35 (2013) A1417–A1439. Crossref, Web of Science, Google Scholar
    • 4. D. Arnold , Finite Element Exterior Calculus (SIAM, 2018). Crossref, Google Scholar
    • 5. D. Arnold, G. Awanou and R. Winther , Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008) 1229–1251. Crossref, Web of Science, Google Scholar
    • 6. D. Arnold and R. Winther , Mixed finite elements for elasticity, Numer. Math. 92 (2002) 401–419. Crossref, Web of Science, Google Scholar
    • 7. D. N. Arnold and K. Hu , Complexes from complexes, Found. Comput. Math. 21 (2021) 1739–1774. Crossref, Google Scholar
    • 8. F. Bassi, L. Botti, A. Colombo, D. A. Di Pietro and P. Tesini , On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations, J. Comput. Phys. 231 (2012) 45–65. Crossref, Web of Science, Google Scholar
    • 9. L. Beir ao da Veiga, F. Brezzi, F. Dassi, L. D. Marini and A. Russo , Serendipity virtual elements for general elliptic equations in three dimensions, Chin. Ann. Math. Ser. B 39 (2018) 315–334. Crossref, Web of Science, Google Scholar
    • 10. L. Beir ao da Veiga and L. Mascotto , Stability and interpolation properties of serendipity nodal virtual elements, Appl. Math. Lett. 142 (2023) 108639. Crossref, Google Scholar
    • 11. D. Boffi, F. Brezzi and M. Fortin , Mixed finite element methods and applications, Springer Series in Computational Mathematics, Vol. 44 (Springer, 2013). Crossref, Google Scholar
    • 12. F. Bonaldi, D. A. Di Pietro, J. Droniou and K. Hu, An exterior calculus framework for polytopal methods, preprint (2023), arXiv:2303.11093. Google Scholar
    • 13. M. Botti, D. A. Di Pietro and M. Salah , A serendipity fully discrete div-div complex on polygonal meshes, C. R. Méc. 351 (2023) 219–249. Crossref, Google Scholar
    • 14. A. Chen and N. Sukumar , Stabilization-free serendipity virtual element method for plane elasticity, Comput. Methods Appl. Mech. Eng. 404 (2023) 115784. Crossref, Web of Science, Google Scholar
    • 15. L. Chen and X. Huang , Discrete hessian complexes in three dimensions, in The Virtual Element Method and its Applications (Springer, 2020), pp. 93–135. Google Scholar
    • 16. L. Chen and X. Huang , Finite elements for div- and divdiv-conforming symmetric tensors in arbitrary dimension, SIAM J. Numer. Anal. 60 (2022) 1932–1961. Crossref, Web of Science, Google Scholar
    • 17. L. Chen and X. Huang , Finite elements for divdiv-conforming symmetric tensors in three dimensions, Math. Comput. 91 (2022) 1107–1142. Google Scholar
    • 18. L. Chen and X. Huang, Complexes from complexes: Finite element complexes in three dimensions, preprint (2023), arXiv:2211.08656v2. Google Scholar
    • 19. E. B. Chin, J. B. Lasserre and N. Sukumar , Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra, Comput. Mech. 56 (2015) 967–981. Crossref, Web of Science, Google Scholar
    • 20. S. H. Christiansen, J. Gopalakrishnan, J. Guzmán and K. Hu , A discrete elasticity complex on three-dimensional Alfeld splits, Numer. Math. 156 (2024) 159–204. Crossref, Google Scholar
    • 21. D. A. Di Pietro and J. Droniou , Ws,p-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray–Lions problems, Math. Models Methods Appl. Sci. 27 (2017) 879–908. Link, Web of Science, Google Scholar
    • 22. D. A. Di Pietro and J. Droniou , A third Strang lemma for schemes in fully discrete formulation, Calcolo 55 (2018) 40. Crossref, Web of Science, Google Scholar
    • 23. D. A. Di Pietro and J. Droniou , The Hybrid High-Order Method for Polytopal Meshes, Modeling, Simulation and Application, Vol. 19 (Springer International Publishing, 2020). Crossref, Google Scholar
    • 24. D. A. Di Pietro and J. Droniou , An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency, Found. Comput. Math. 23 (2023) 85–164. Crossref, Web of Science, Google Scholar
    • 25. D. A. Di Pietro and J. Droniou , A fully discrete plates complex on polygonal meshes with application to the Kirchhoff–Love problem, Math. Comput. 92 (2023) 51–77. Crossref, Web of Science, Google Scholar
    • 26. D. A. Di Pietro and J. Droniou , Homological- and analytical-preserving serendipity framework for polytopal complexes, with application to the DDR method, ESAIM: Math. Model Numer. Anal. 57 (2023) 191–225. Crossref, Google Scholar
    • 27. D. A. Di Pietro, J. Droniou and S. Pitassi , Cohomology of the discrete de Rham complex on domains of general topology, Calcolo 60 (2023) 32. Crossref, Google Scholar
    • 28. D. A. Di Pietro, J. Droniou and F. Rapetti , Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra, Math. Models Methods Appl. Sci. 30 (2020) 1809–1855. Link, Web of Science, Google Scholar
    • 29. J. Hu, Y. Liang and R. Ma , Conforming finite element divdiv complexes and the application for the linearized Einstein–Bianchi system, SIAM J. Numer. Anal. 60 (2022) 1307–1330. Crossref, Google Scholar
    • 30. J. Hu, Y. Liang, R. Ma and M. Zhang, New conforming finite element divdiv complexes in three dimensions, preprint (2022), arXiv:2204.07895. Google Scholar
    • 31. D. Pauly and W. Zulehner , The divDiv-complex and applications to biharmonic equations, Appl. Anal. 99 (2020) 1579–1630. Crossref, Google Scholar
    Remember to check out the Most Cited Articles!

    View our Mathematical Modelling books
    Featuring authors Frederic Y M Wan, Gregory Baker and more!