Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

HYBRIDIZABLE DISCONTINUOUS GALERKIN METHOD FOR ELLIPTIC EQUATIONS WITH NONLINEAR COEFFICIENTS

  • MINAM, MOON (DEPARTMENT OF MATHEMATICS, KOREA MILITARY ACADEMY)
  • Received : 2022.07.12
  • Accepted : 2022.12.12
  • Published : 2022.12.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we analyze the hybridizable discontinuous Galerkin (HDG) method for second-order elliptic equations with nonlinear coefficients, which are used in many fields. We present the HDG method that uses a mixed formulation based on numerical trace and flux. Under assumptions on the nonlinear coefficient and H2-regularity for a dual problem, we prove that the discrete systems are well-posed and the numerical solutions have the optimal order of convergence as a mesh parameter. Also, we provide a matrix formulation that can be calculated using an iterative technique for numerical experiments. Finally, we present representative numerical examples in 2D to verify the validity of the proof of Theorem 3.10.

Keywords

Acknowledgement

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. NRF-2020R1F1A1A01072414, NRF-2022R1F1A1069217).

References

  1. M. Moon. Generalized multiscale hybridizable discontinuous Galerkin (GMsHDG) method for flows in nonlinear porous media, Journal of Computational and Applied Mathematics, 415 (2022), 114440.
  2. B. Cockburn, J. Gopalakrishnan, and R. Lazarov. Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM Journal of Numerical Analysis, 47 (2009), 1319-1365. https://doi.org/10.1137/070706616
  3. D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM Journal of Numerical Analysis, 39 (2002), 1749-1779. https://doi.org/10.1137/S0036142901384162
  4. R. Kirby, S. Sherwin, and B. Cockburn. To CG or to HDG : A comparative study, Journal of Scientific Computing, 51 (2011), 183-212.
  5. A. Huerta, A. Angeloski, X. Roca, and J. Peraire. Efficiency of high-order elements for continuous and discontinuous Galerkin methods, International Journal for Numerical Methods in Engineering, 96 (2013), 529-560. https://doi.org/10.1002/nme.4547
  6. G. Giorgiani, S. Fernandez-Mendez, and A. Huerta. Hybridizable discontinuous Galerkin p-adaptivity for wave propagation problems, International Journal for Numerical Methods in Fluids, 72 (2013), 1244-1262. https://doi.org/10.1002/fld.3784
  7. G. Giorgiani, S. Fernandez-Mendez, and A. Huerta. Hybridizable discontinuous Galerkin with degree adaptivity for the incompressible Navier-Stokes equations, Computers & Fluids, 98 (2014), 196-208. https://doi.org/10.1016/j.compfluid.2014.01.011
  8. S. Yakovlev, D. Moxey, R. M. Kirby, and S. J. Sherwin. To CG or to HDG : A comparative study in 3D, Journal of Scientific Computing, 67 (2016), 192-220. https://doi.org/10.1007/s10915-015-0076-6
  9. B. Cockburn, W. Qiu, and K. Shi. Conditions for superconvergence of HDG methods for second-order elliptic problems, Mathematics of Computation, 81 (2012), 1327-1353. https://doi.org/10.1090/S0025-5718-2011-02550-0
  10. B. Cockburn, J. Gopalakrishnan, and F-J. Sayas, A projection-based error analysis of HDG methods, Mathematics of Computation, 79 (2010), 1351-1367. https://doi.org/10.1090/S0025-5718-10-02334-3
  11. N.C. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, Journal of Computational Physics, 228 (2009), 3232-3254. https://doi.org/10.1016/j.jcp.2009.01.030
  12. N.C. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations, Journal of Computational Physics, 228 (2009), 8841-8855. https://doi.org/10.1016/j.jcp.2009.08.030
  13. M. Moon, H.K. Jun, and T. Suh, Error estimates on hybridizable discontinuous Galerkin methods for parabolic equations with nonlinear coefficients, Advances in Mathematical Physics, 17 (2017), 1-11. https://doi.org/10.1155/2017/9736818
  14. M. Stanglmeier, N.C. Nguyen, J. Peraire, and B. Cockburn. An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation, Computer Methods in Applied Mechanics and Engineering, 300 (2016), 748-769. https://doi.org/10.1016/j.cma.2015.12.003
  15. M, Kronbichler, S. Schoeder, C. Muller, and W.A. Wall. Comparison of implicit and explicit hybridizable discontinuous Galerkin methods for the acoustic wave equation, International Journal for Numerical Methods in Engineering, 106 (2016), 712-739. https://doi.org/10.1002/nme.5137
  16. N.C. Nguyen, J. Peraire, and B. Cockburn. A hybridizable discontinuous Galerkin method for Stokes flow, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 582-597. https://doi.org/10.1016/j.cma.2009.10.007
  17. G.N. Gatica and F.A. Sequeira. Analysis of an augmented HDG method for a class of quasi-Newtonian Stokes flows, Journal of Scientific Computing, 65 (2015), 1270-1308. https://doi.org/10.1007/s10915-015-0008-5
  18. J. Peraire, N.C. Nguyen, and B. Cockburn. A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations, AIAA, 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, Florida 2010.
  19. N.C. Nguyen, J. Peraire, and B. Cockburn. A hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations, 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition. 2010.
  20. N.C. Nguyen, J. Peraire, and B. Cockburn. An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations, Journal of Computational Physics, 230 (2011), 1147-1170. https://doi.org/10.1016/j.jcp.2010.10.032
  21. A. Cesmelioglu, B. Cockburn, and W. Qiu. Analysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier-Stokes equations, Mathematics of Computation, 86 (2017), 1643-1670. https://doi.org/10.1090/mcom/3195
  22. E.-J. Park. Mixed finite element methods for nonlinear second-order elliptic problems, SIAM Journal on Numerical Analysis, 32 (1995), 865-885. https://doi.org/10.1137/0732040
  23. D. Kim and E.-J. Park. A priori and a posteriori analysis of mixed finite element methods for nonlinear elliptic equations, SIAM Journal on Numerical Analysis, 48 (2010), 1186-1207. https://doi.org/10.1137/090747002
  24. Y. Sangita, A. Pani, and E.-J. Park. Superconvergent discontinuous Galerkin methods for nonlinear elliptic equations, Mathematics of Computation, 82 (2013), 1297-1335. https://doi.org/10.1090/S0025-5718-2013-02662-2
  25. A. Muhammad, E.-J. Park, and D. Shin. Analysis of multiscale mortar mixed approximation of nonlinear elliptic equations, Computers and Mathematics with Applications, 75 (2018), 401-418. https://doi.org/10.1016/j.camwa.2017.09.031
  26. R. Sevilla and A. Huerta. Tutorial on Hybridizable Discontinuous Galerkin (HDG) for second-order elliptic problems, Advanced finite element technologies 566, Springer, Cham, 2016.
  27. M. Moon and Y. H. Lim. Superconvergence of Hybridizable Discontinuous Galerkin method for second-order elliptic equations, Journal of the Korean Society for Industrial and Applied Mathematics, 20 (2016), 295-308. https://doi.org/10.12941/jksiam.2016.20.295
  28. P. Grisvard. Elliptic problems in nonsmooth domains, Classics in Applied Mathematics 69, Pitman, Boston, MA, 1985.
  29. P. Knabner. Numerical methods for elliptic and parabolic partial differential equations, Springer, New York, 2003.
  30. L.C. Evans. Partial differential equations, Graduate Studies on Mathematics 19, American Mathematical Society, 2010.