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

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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.

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References

  1. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Elsevier, Amsterdam (2003)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Banks, J., Hagstrom, T.: On Galerkin difference methods. J. Comput. Phys. 313, 310–327 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, vol. 15, 3rd edn. Springer, New York (2008)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Book  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  16. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  17. Clément, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9, 77–84 (1975)

    MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  21. Dekel, S., Leviatan, D.: The Bramble–Hilbert lemma for convex domains. SIAM J. Math. Anal. 35, 1203–1212 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  22. Di Pierto, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69. Springer, Berlin (2012)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  24. Dupont, T., Scott, L.R.: Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34, 441–463 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  25. Georgoulis, E., Pryer, T.: Recovered finite element methods. Comput. Methods Appl. Mech. Eng. 332, 303–324 (2018)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  27. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  30. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, p. 233. Springer, Berlin (2008)

    Book  MATH  Google Scholar 

  31. Larsson, K., Larson, M.G.: Continuous piecewise linear finite elements for the Kirchhoff–Love plate equation. Numer. Math. 121, 65–97 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  40. Nay, R.A., Utku, S.: An alternative for the finite element method. In: Variational Methods in Engineering, vol. 1. University of Southampton (1972)

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

  44. Reichel, L.: On polynomial approximation in the uniform norm by the discrete least squares method. BIT 26, 349–368 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sukumar, N., Tabarraei, A.: Conforming polygonal finite elements. Int. J. Numer. Methods Eng. 61, 2045–2066 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  46. Sullivan, D.O.: Exploring spatial process dynamics using irregular cellular automaton models. Geogr. Anal. 33, 1–18 (2001)

    Article  Google Scholar 

  47. Venkatakrishnan, V.: Convergence to steady state solutions of the Euler equations on unstructured grids with limiters. J. Comput. Phys. 118, 120–130 (1995)

    Article  MATH  Google Scholar 

  48. Wendland, H.: Scattered Data Approximation. Cambridge Univeristy Press, Cambridge (2005)

    MATH  Google Scholar 

  49. Wilhelmsen, D.: A Markov inequality in several dimensions. J. Approx. Theory 11, 216–220 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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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)

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Authors and Affiliations

  1. CAPT, LMAM and School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Ruo Li

  2. LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, No. 55, Zhong-Guan-Cun East Road, Beijing, 100190, People’s Republic of China

    Pingbing Ming

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, No. 19A, Yu-Quan Road, Beijing, 100049, People’s Republic of China

    Pingbing Ming

  4. School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China

    Ziyuan Sun

  5. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, People’s Republic of China

    Zhijian Yang

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  1. Ruo Li
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Correspondence to Pingbing Ming.

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Li, R., Ming, P., Sun, Z. et al. An Arbitrary-Order Discontinuous Galerkin Method with One Unknown Per Element. J Sci Comput 80, 268–288 (2019). https://doi.org/10.1007/s10915-019-00937-y

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