Nonstandard finite element de Rham complexes on cubical meshes (original) (raw)

References

  1. Adini, A., Clough, R.W.: Analysis of plate bending by the finite element method. NSF Report G. 7337, University of California, Berkeley (1961)
  2. Arnold, D., Awanou, G.: Finite element differential forms on cubical meshes. Math. Comput. 83(288), 1551–1570 (2014)
    MathSciNet MATH Google Scholar
  3. Arnold, D.N., Awanou, G.: The serendipity family of finite elements. Found. Comput. Math. 11(3), 337–344 (2011)
    MathSciNet MATH Google Scholar
  4. Arnold, D.N., Boffi, D., Bonizzoni, F.: Finite element differential forms on curvilinear cubic meshes and their approximation properties. Numer. Math. 129(1), 1–20 (2015)
    MathSciNet MATH Google Scholar
  5. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1 (2006)
    MathSciNet MATH Google Scholar
  6. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47(2), 281–354 (2010)
    MathSciNet MATH Google Scholar
  7. Arnold, D.N., Logg, A.: Periodic table of the finite elements. SIAM News 47(9), 212 (2014)
    Google Scholar
  8. Arnold, D.N., Qin, J.: Quadratic velocity/linear pressure Stokes elements. Adv. Comput. Methods Partial Differ. Equ. 7, 28–34 (1992)
    Google Scholar
  9. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)
    MATH Google Scholar
  10. Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology, vol. 82. Springer, Berlin (2013)
    MATH Google Scholar
  11. Brenner, S.C.: Forty years of the Crouzeix–Raviart element. Numer. Methods Partial Differ. Equ. 31(2), 367–396 (2015)
    MathSciNet MATH Google Scholar
  12. Brenner, S.C., Scott, R.: The Mathematical Theory of Finite Element Methods, vol. 15. Springer, Berlin (2007)
    Google Scholar
  13. Brezzi, F., Douglas Jr., J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217–235 (1985)
    MathSciNet MATH Google Scholar
  14. Buffa, A., Rivas, J., Sangalli, G., Vázquez, R.: Isogeometric discrete differential forms in three dimensions. SIAM J. Numer. Anal. 49(2), 818–844 (2011)
    MathSciNet MATH Google Scholar
  15. Christiansen, S.H., Hu, J., Hu, K.: Nodal finite element de Rham complexes. Numer. Math. 139, 1–36 (2016)
    MathSciNet MATH Google Scholar
  16. Christiansen, S.H., Hu, K.: Generalized finite element systems for smooth differential forms and Stokes’ problem. Numerische Mathematik 140(2), 327–371 (2018)
    MathSciNet MATH Google Scholar
  17. Christiansen, S.H., Munthe-Kaas, H.Z., Owren, B.: Topics in structure-preserving discretization. Acta Numer. 20, 1–119 (2011)
    MathSciNet MATH Google Scholar
  18. Da Veiga, L.B., Brezzi, F., Marini, L., Russo, A.: Serendipity nodal VEM spaces. Comput. Fluids 141, 2–12 (2016)
    MathSciNet MATH Google Scholar
  19. Da Veiga, L.B., Brezzi, F., Marini, L.D., Russo, A.: Serendipity face and edge VEM spaces. Preprint arXiv:1606.01048 (2016)
  20. Dummit, D.S., Foote, R.M.: Abstract Algebra, vol. 3. Wiley, Hoboken (2004)
    MATH Google Scholar
  21. Falk, R.S., Neilan, M.: Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51(2), 1308–1326 (2013)
    MathSciNet MATH Google Scholar
  22. Floater, M., Gillette, A.: Nodal bases for the serendipity family of finite elements. Found. Comput. Math. 17(4), 879–893 (2017)
    MathSciNet MATH Google Scholar
  23. Gillette, A., Kloefkorn, T.: Trimmed serendipity finite element differential forms. Math. Comput. 88(316), 583–606 (2019)
    MathSciNet MATH Google Scholar
  24. Hiptmair, R.: Canonical construction of finite elements. Math. Comput. Am. Math. Soc. 68(228), 1325–1346 (1999)
    MathSciNet MATH Google Scholar
  25. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11(July 2003), 237–339 (2002)
    MathSciNet MATH Google Scholar
  26. Hu, J.: Finite element approximations of symmetric tensors on simplicial grids in \(\mathbb{R}^{n}\): the higher order case. J. Comput. Math. 33, 283–296 (2015)
    MathSciNet MATH Google Scholar
  27. Hu, J., Yang, X., Zhang, S.: Capacity of the Adini element for biharmonic equations. J. Sci. Comput. 69(3), 1366–1383 (2016)
    MathSciNet MATH Google Scholar
  28. Hu, J., Zhang, S.: A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids. Sci. China Math. 58(2), 297–307 (2015)
    MathSciNet MATH Google Scholar
  29. John, V., Linke, A., Merdon, C., Neilan, M., Rebholz, L.G.: On the divergence constraint in mixed finite element methods for incompressible flows. SIAM Rev. 59(3), 492–544 (2017)
    MathSciNet MATH Google Scholar
  30. Linke, A., Merdon, C., Neilan, M., Neumann, F.: Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes-problem. Math. Comput. 87, 1543–1566 (2018)
    MathSciNet MATH Google Scholar
  31. Mardal, K.A., Tai, X.C., Winther, R.: A robust finite element method for Darcy–Stokes flow. SIAM J. Numer. Anal. 40(5), 1605–1631 (2002)
    MathSciNet MATH Google Scholar
  32. Nédélec, J.C.: Mixed finite elements in \(\mathbb{R}^{3}\). Numer. Math. 35(3), 315–341 (1980)
    MathSciNet MATH Google Scholar
  33. Nédélec, J.C.: A new family of mixed finite elements in \(\mathbb{R}^{3}\). Numer. Math. 50(1), 57–81 (1986)
    MathSciNet MATH Google Scholar
  34. Neilan, M.: Discrete and conforming smooth de Rham complexes in three dimensions. Math. Comput. 84, 2059–2081 (2015)
    MathSciNet MATH Google Scholar
  35. Neilan, M., Sap, D.: Stokes elements on cubic meshes yielding divergence-free approximations. Calcolo 53(3), 263–283 (2016)
    MathSciNet MATH Google Scholar
  36. Nilssen, T., Tai, X.C., Winther, R.: A robust nonconforming \(H^{2}\)-element. Math. Comput. 70(234), 489–505 (2001)
    MATH Google Scholar
  37. Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Dold, A., Eckmann, B. (eds.) Mathematical Aspects of Finite Element Methods. Lecture Notes in Mathematics, vol. 606. Springer, Berlin, Heidelberg (1977). https://doi.org/10.1007/BFb0064470
  38. Stenberg, R.: A nonstandard mixed finite element family. Numer. Math. 115(1), 131–139 (2010)
    MathSciNet MATH Google Scholar
  39. Tai, X.C., Winther, R.: A discrete de Rham complex with enhanced smoothness. Calcolo 43(4), 287–306 (2006)
    MathSciNet MATH Google Scholar
  40. Wang, M.: The generalized Korn inequality on nonconforming finite element spaces. Chin. J. Numer. Math. Appl. 16, 91–96 (1994)
    MathSciNet MATH Google Scholar
  41. Wang, M.: On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements. SIAM J. Numer. Anal. 39(2), 363–384 (2001)
    MathSciNet MATH Google Scholar
  42. Wang, M., Shi, Z.C., Xu, J.: A new class of Zienkiewicz-type non-conforming element in any dimensions. Numer. Math. 106(2), 335–347 (2007)
    MathSciNet MATH Google Scholar
  43. Zhang, S.: A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comput. 74(250), 543–554 (2005)
    MathSciNet MATH Google Scholar
  44. Zhang, S.: Stable finite element pair for Stokes problem and discrete Stokes complex on quadrilateral grids. Numer. Math. 133(2), 371–408 (2016)
    MathSciNet MATH Google Scholar

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