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An Arbitrary-Order Discontinuous Galerkin Method with One Unknown Per Element

  • Ruo Li
  • Pingbing MingEmail author
  • Ziyuan Sun
  • Zhijian Yang
Article
  • 34 Downloads

Abstract

We propose an arbitrary-order discontinuous Galerkin method for second-order elliptic problem on general polygonal mesh with only one degree of freedom per element. This is achieved by locally solving a discrete least-squares over a neighboring element patch. Under a geometrical condition on the element patch, we prove an optimal a priori error estimates in the energy norm and in the \(\hbox {L}^2\) norm. The accuracy and the efficiency of the method up to order six on several polygonal meshes are illustrated by a set of benchmark problems.

Keywords

Least-squares reconstruction Discontinuous Galerkin method Elliptic problem 

Mathematics Subject Classification

Primary 65N30 49N45 Secondary 74K20 

Notes

Acknowledgements

The authors would like to thank Dr. Fengyang Tang for his help in the earlier stage of the present work, and the authors would like to thank the anonymous referees for the constructive comments that improve the paper. Funding was provided by National Natural Science Foundation of China (Grant Nos. 11425106, 91630313, 91630313, 11671312)

References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Elsevier, Amsterdam (2003)zbMATHGoogle Scholar
  2. 2.
    Antonietti, P.F., Giani, S., Houston, P.: \(hp\)-version composite discontinuous Galerkin methods for elliptic problems on complicated domains. SIAM J. Sci. Comput. 35, A1417–A1439 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Banks, J., Hagstrom, T.: On Galerkin difference methods. J. Comput. Phys. 313, 310–327 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barth, T.J., Larson, M.G.: A posteriori error estimates for higher order Godunov finite volume methods on unstructured meshes. Finite volumes for complex applications, III (Porquerolles, 2002), pp. 27–49. Hermes Sci. Publ, Paris (2002)Google Scholar
  7. 7.
    Bassi, F., Botti, L., Colombo, A., Rebay, S.: Agglomeration-based discontinuous galerkin discretization of Euler and Navier–Stokes equations. Comput. Fluids 61, 77–85 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bassi, F., Botti, L., Colombo, A.: Agglomeration-based physical frame dG discretization: an attempt to be mesh free. Math. Models Methods Appl. Sci. 24, 1495–1539 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, vol. 15, 3rd edn. Springer, New York (2008)CrossRefzbMATHGoogle Scholar
  10. 10.
    Brezzi, F., Lipnikov, K., Simoncini, V.: A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 15, 1533–1551 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Brezzi, F., Buffa, A., Lipnikov, K.: Mimetic finite differences for elliptic problems. ESAIM Numer. Anal. 43, 277–295 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cangiani, A., Georgoulis, E.H., Houston, P.: \(hp\)-version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci. 24, 2009–2041 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cangiani, A., Dong, Z.N., Georgoulis, E.H., Houston, P.: hp-version discontinuous galerkin methods for advection–diffusion–reaction problems on polytopic meshes. ESAIM Math. Model. Numer. Anal. 50, 699–725 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cangiani, A., Dong, Z.N., Georgoulis, E.H., Houston, P.: hp-Version Discontinuous Galerkin Methods on Polygonal and Polyhedral Meshes. SpringerBriefs in Mathematics. Springer, Berlin (2017)CrossRefzbMATHGoogle Scholar
  15. 15.
    Chen, H.T., Guo, H.L., Zhang, Z.M., Zou, Q.S.: A \({\rm C}^{0}\) linear finite element method for two fourth order eigenvalue problems. IAM J. Numer. Anal. 37, 2120–2138 (2017)zbMATHGoogle Scholar
  16. 16.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  17. 17.
    Clément, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9, 77–84 (1975)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Cockburn, B., Karniadakis, G.E., Shu, C.W.: The development of discontinuous Galerkin methods. In: Newport, R.I. (ed.) Discontinuous Galerkin Methods, Lect. Notes Comput. Sci. Eng., vol. 11, pp. 3–50. Springer, Berlin (1999)CrossRefGoogle Scholar
  19. 19.
    Cockburn, B., Gopalakrishnan, J., Lazarov, R.: Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47, 1319–1365 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    da Veiga, L.B., Lipnikov, K., Manzini, G.: The Mimetic Finite Difference Method for Elliptic Problems, MS&A. Modeling, Simulation and Applications, vol. 11. Springer, Cham (2014)zbMATHGoogle Scholar
  21. 21.
    Dekel, S., Leviatan, D.: The Bramble–Hilbert lemma for convex domains. SIAM J. Math. Anal. 35, 1203–1212 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Di Pierto, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69. Springer, Berlin (2012)zbMATHGoogle Scholar
  23. 23.
    Di Pierto, D.A., Ern, A., Lemaire, S.: An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Comput. Methods Appl. Math. 14, 461–472 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Dupont, T., Scott, L.R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34, 441–463 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Georgoulis, E., Pryer, T.: Recovered finite element methods. Comput. Methods Appl. Mech. Eng. 332, 303–324 (2018)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Geuzaine, C., Remacle, J.F.: Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79, 1309–1331 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)zbMATHGoogle Scholar
  28. 28.
    Guo, H.L., Zhang, Z.M., Zou, Q.S.: A \({\rm C}^{0}\) linear finite element method for biharmonic problems. J. Sci. Comput. 74, 1397–1422 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hampshire, J.K., TBH, V., CH, C.: Three node triangular bending elements with one degree of freedom per node. Eng. Comput. 9(1), 49–62 (1992)CrossRefGoogle Scholar
  30. 30.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, p. 233. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  31. 31.
    Larsson, K., Larson, M.G.: Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation. Numer. Math. 121, 65–97 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Li, R., Ming, P.B., Tang, F.Y.: An efficient high order heterogeneous multiscale method for elliptic problems. Multiscale Model. Simul. 10, 259–283 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Li, R., Ming, P.B., Sun, Z.Y., Yang, F.Y., Yang, Z.J.: A discontinuous Galerkin method by patch reconstruction for biharmonic problem. J. Comput. Math. 37, 561–578 (2019a)CrossRefGoogle Scholar
  34. 34.
    Li, R., Sun, Z.Y., Yang, F.Y.: Solving eigenvalue problems in a discontinuous Galerkin approximate space by patch reconstruction (2019b). arXiv:1901.01803
  35. 35.
    Li, R., Sun, Z.Y., Yang, F.Y., Yang, Z.J.: A finite element method by patch reconstruction for the Stokes problem using mixed formulations. J. Comput. Appl. Math. 353, 1–20 (2019c)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Lipnikov, K., Vassilev, D., Yotov, I.: Discontinuous Galerkin and mimetic finite difference methods for coupled Stokes–Darcy flows on polygonal and polyhedral grids. Numer. Math. 126, 1–40 (2013)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Mozolevski, I., Süli, E.: A priori error analysis for the \(hp\)-version of the discontinuous Galerkin finite element method for the biharmonic equation. Comput. Methods Appl. Math. 3, 596–607 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Mu, L., Wang, J.P., Wang, Y.Q., Ye, X.: Interior penalty discontinuous Galerkin method on very general polygonal and polyhedral meshes. J. Comput. Appl. Math. 255, 432–440 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Narcowich, F.J., Ward, J.D., Wendland, H.: Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting. Math. Comput. 74, 743–763 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Nay, R.A., Utku, S.: An alternative for the finite element method. In: Variational Methods in Engineering, vol. 1. University of Southampton (1972)Google Scholar
  41. 41.
    Oñate, E., Cervera, M.: Derivation of thin plate bending elements with one degree of freedom per node: a simple three node triangle. Eng. Comput. 10, 543–561 (1993)CrossRefGoogle Scholar
  42. 42.
    Phaal, R., Calladine, C.R.: A simple class of finite elements for plate and shell problems. II: an element for thin shells, with only translational degrees of freedom. Int. J. Numer. Methods Eng. 35, 979–996 (1992)CrossRefzbMATHGoogle Scholar
  43. 43.
    Reed, W.H., Hill, T.R.: Triangular Mesh Methods for the Neutron Transport Equation. Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973)Google Scholar
  44. 44.
    Reichel, L.: On polynomial approximation in the uniform norm by the discrete least squares method. BIT 26, 349–368 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Sukumar, N., Tabarraei, A.: Conforming polygonal finite elements. Int. J. Numer. Methods Eng. 61, 2045–2066 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Sullivan, D.O.: Exploring spatial process dynamics using irregular cellular automaton models. Geogr. Anal. 33, 1–18 (2001)CrossRefGoogle Scholar
  47. 47.
    Venkatakrishnan, V.: Convergence to steady state solutions of the Euler equations on unstructured grids with limiters. J. Comput. Phys. 118, 120–130 (1995)CrossRefzbMATHGoogle Scholar
  48. 48.
    Wendland, H.: Scattered Data Approximation. Cambridge Univeristy Press, Cambridge (2005)zbMATHGoogle Scholar
  49. 49.
    Wilhelmsen, D.: A Markov inequality in several dimensions. J. Approx. Theory 11, 216–220 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Wirasaet, D., Kubatko, E.J., Michoski, C.E., Tanaka, S., Westerink, J.J., Dawson, C.: Discontinuous Galerkin methods with nodal and hybrid modal/nodal triangular, quadrilateral, and polygonal elements for nonlinear shallow water flow. Comput. Methods Appl. Mech. Eng. 270, 113–149 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Yamakawa, S., Shimada, K.: Converting a tetrahedral mesh to a prism-tetrahedral hybrid mesh for FEM accuracy and efficiency. Int. J. Numer. Methods Eng. 80, 2099–2129 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Ruo Li
    • 1
  • Pingbing Ming
    • 2
    • 3
    Email author
  • Ziyuan Sun
    • 4
  • Zhijian Yang
    • 5
  1. 1.CAPT, LMAM and School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China
  2. 2.LSEC, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China
  4. 4.School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China
  5. 5.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China

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