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\(H({\text {div}})\) and \(H(\mathbf{curl})\)-conforming virtual element methods

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Abstract

In the present paper we construct virtual element spaces that are \(H(\mathrm{div})\)-conforming and \(H(\mathbf{curl})\)-conforming on general polygonal and polyhedral elements; these spaces can be interpreted as a generalization of well known finite elements. We moreover present the basic tools needed to make use of these spaces in the approximation of partial differential equations. Finally, we discuss the construction of exact sequences of VEM spaces.

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References

  1. Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66(3), 376–391 (2013)

    Article  MathSciNet  Google Scholar 

  2. Arnold, D., Awanou, G., Boffi, D., Bonizzoni, F., Falk, R., Winther, R.: The periodic table of finite elements (2015)

  3. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arroyo, M., Ortiz, M.: Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. Int. J. Numer. Methods Eng. 65(13), 2167–2202 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Auchmuty, G., Alexander, J.C.: \(L^2\) well-posedness of planar div-curl systems. Arch. Ration. Mech. Anal. 160(2), 91–134 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Auchmuty, G., Alexander, J.C.: \(L^2\)-well-posedness of 3D div-curl boundary value problems. Q. Appl. Math. 63(3), 479–508 (2005)

    Article  MathSciNet  Google Scholar 

  7. Babuška, I., Banerjee, U., Osborn, J.E.: Survey of meshless and generalized finite element methods: a unified approach. Acta Numer. 12, 1–125 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Babuška, I., Banerjee, U., Osborn, J.E.: Generalized finite element methods—main ideas, results and perspective. Int. J. Comput. Methods 01(01), 67–103 (2004)

    Article  MATH  Google Scholar 

  9. Babuška, I., Melenk, J.M.: The partition of unity method. Int. J. Numer. Methods Eng. 40(4), 727–758 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)

  11. Beirão da Veiga, L., Brezzi, F., Marini, L.D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)

  12. Beirão da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: The hitchhiker’s guide to the virtual element method. Math. Models Methods Appl. Sci. 24(8), 1541–1573 (2014)

  13. Beirão da Veiga, L., Lipnikov, K., Manzini, G.: Convergence analysis of the high-order mimetic finite difference method. Numer. Math. 113(3), 325–356 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Beirão da Veiga, L., Lipnikov, K., Manzini, G.: Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49(5), 1737–1760 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Beirão da Veiga, L., Lipnikov, K., Manzini, G.: The Mimetic Finite Difference Method for Elliptic Problems. Modeling, Simulation and Applications, vol. 11. Springer, Berlin (2014)

  16. Beirão da Veiga, L., Manzini, G.: A higher-order formulation of the mimetic finite difference method. SIAM J. Sci. Comput. 31(1), 732–760 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Beirão da Veiga, L., Manzini, G.: A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34(2), 759–781 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Bishop, J.E.: A displacement-based finite element formulation for general polyhedra using harmonic shape functions. Int. J. Numer. Methods Eng. 97(1), 1–31 (2014)

    Article  MathSciNet  Google Scholar 

  19. Bochev, P.B., Hyman, J.M.: Principles of mimetic discretizations of differential operators. In: Compatible Spatial Discretizations. IMA Volumes in Mathematics and Its Applications, vol. 142, pp. 89–119. Springer, New York (2006)

  20. Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  21. Bonelle, J., Ern, A.: Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes. ESAIM Math. Model. Numer. Anal. 48(2), 553–581 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bossavit, A.: Mixed finite elements and the complex of Whitney forms. In: The Mathematics of Finite Elements and Applications. VI (Uxbridge, 1987), pp. 137–144. Academic Press, London (1988)

  23. Brezzi, F., Buffa, A., Lipnikov, K.: Mimetic finite differences for elliptic problems. Math. Model. Numer. Anal. 43(2), 277–295 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Brezzi, F., Falk, R.S., Marini, L.D.: Basic principles of mixed virtual element methods. ESAIM Math. Model. Numer. Anal. 48(4), 1227–1240 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of mimetic finite difference methods for diffusion problems on polyhedral meshes. SIAM J. Num. Anal. 43, 1872–1896 (2005)

  26. Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of mimetic finite difference method for diffusion problems on polyhedral meshes with curved faces. Math. Models Methods Appl. Sci. 16(2), 275–297 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Brezzi, F., Lipnikov, K., Shashkov, M., Simoncini, V.: A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput. Methods Appl. Mech. Eng. 196(37–40), 3682–3692 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15(10), 1533–1551 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  29. Brezzi, F., Marini, L.D.: Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Eng. 253, 455–462 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Di Pietro, D., Alexandre Ern, A.: A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Methods Appl. Mech. Eng. 283, 1–21 (2015)

    Article  MathSciNet  Google Scholar 

  31. Di Pietro, D., Ern, A.: A family of arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes. https://hal.archives-ouvertes.fr/hal-00918482 (2013)

  32. Droniou, J., Eymard, R., Gallouët, T., Herbin, R.: A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods. Math. Models Methods Appl. Sci. 20(2), 265–295 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Floater, M., Hormann, K., Kós, G.: A general construction of barycentric coordinates over convex polygons. Adv. Comput. Math. 24(1–4), 311–331 (2006)

    Article  MATH  Google Scholar 

  34. Fries, T.P., Belytschko, T.: The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Methods Eng. 84(3), 253–304 (2010)

    MathSciNet  MATH  Google Scholar 

  35. Griebel, M., Schweitzer, M.A.: A particle-partition of unity method for the solution of elliptic, parabolic, and hyperbolic PDEs. SIAM J. Sci. Comput. 22(3), 853–890 (2000, electronic)

  36. Griebel, M., Schweitzer, M.A.: A particle-partition of unity method. II. Efficient cover construction and reliable integration. SIAM J. Sci. Comput. 23(5), 1655–1682 (2002, electronic)

  37. Gyrya, V., Lipnikov, K.: High-order mimetic finite difference method for diffusion problems on polygonal meshes. J. Comput. Phys. 227(20), 8841–8854 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Hiptmair, R.: Discrete Hodge operators. Numer. Math. 90(2), 265–289 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hyman, J.M., Shashkov, M.: Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids. Appl. Numer. Math. 25(4), 413–442 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kuznetsov, Y., Repin, S.: New mixed finite element method on polygonal and polyhedral meshes. Russ. J. Numer. Anal. Math. Model. 18(3), 261–278 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Lipnikov, K., Manzini, G., Shashkov, M.: Mimetic finite difference method. J. Comput. Phys. 257(part B), 1163–1227 (2014)

  42. Martin, S., Kaufmann, P., Botsch, M., Wicke, M., Gross, M.: Polyhedral finite elements using harmonic basis functions. Comput. Graph. Forum 27(5), 1521–1529 (2008)

    Article  Google Scholar 

  43. Mattiussi, C.: An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology. J. Comput. Phys. 133(2), 289–309 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  44. Sukumar, N., Malsch, E.A.: Recent advances in the construction of polygonal finite element interpolants. Arch. Comput. Methods Eng. 13(1), 129–163 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Tabarraei, A., Sukumar, N.: Extended finite element method on polygonal and quadtree meshes. Comput. Methods Appl. Mech. Eng. 197(5), 425–438 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Talischi, C., Paulino, G.H., Pereira, A., Menezes, I.F.M.: Polygonal finite elements for topology optimization: a unifying paradigm. Int. J. Numer. Methods Eng. 82(6), 671–698 (2010)

    MATH  Google Scholar 

  47. Talischi, C., Paulino, G.H., Pereira, A., Menezes, I.F.M.: PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct. Multidiscip. Optim. 45(3), 309–328 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. Wachspress, E.: A Rational Finite Element Basis. Mathematics in Science and Engineering, vol. 114. Academic Press, Inc., New York (1975)

  49. Wachspress, E.: Rational bases for convex polyhedra. Comput. Math. Appl. 59(6), 1953–1956 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133, Milano, Italy

    L. Beirão da Veiga

  2. IUSS, Piazza della Vittoria 15, 27100, Pavia, Italy

    F. Brezzi

  3. Dipartimento di Matematica, Università di Pavia, Pavia, Italy

    L. D. Marini

  4. Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, via Cozzi 57, 20125, Milano, Italy

    A. Russo

  5. IMATI del CNR, Via Ferrata 1, 27100, Pavia, Italy

    L. Beirão da Veiga, F. Brezzi, L. D. Marini & A. Russo

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  1. L. Beirão da Veiga
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  2. F. Brezzi
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  3. L. D. Marini
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  4. A. Russo
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Correspondence to L. Beirão da Veiga.

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da Veiga, L.B., Brezzi, F., Marini, L.D. et al. \(H({\text {div}})\) and \(H(\mathbf{curl})\)-conforming virtual element methods. Numer. Math. 133, 303–332 (2016). https://doi.org/10.1007/s00211-015-0746-1

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  • DOI: https://doi.org/10.1007/s00211-015-0746-1

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