DOI QR코드

DOI QR Code

SUPERCONVERGENCE OF HYBRIDIZABLE DISCONTINUOUS GALERKIN METHOD FOR SECOND-ORDER ELLIPTIC EQUATIONS

  • MOON, MINAM (DEPARTMENT OF MATHEMATICS, KOREA MILITARY ACADEMY) ;
  • LIM, YANG HWAN (DEPARTMENT OF MATHEMATICS, KOREA MILITARY ACADEMY)
  • 투고 : 2016.09.26
  • 심사 : 2016.11.24
  • 발행 : 2016.12.25

초록

We propose a projection-based analysis of a new hybridizable discontinuous Gale-rkin method for second order elliptic equations. The method is more advantageous than the standard HDG method in a sense that the new method has higher-order accuracy and lower computational cost, and is more flexible. Notable distinctions of our new method, when compared to the standard HDG emthod, are that our method uses $L^2$-projection and suitable stabilization parameter depending on a mesh size for superconvergence. We show that the error for the solution of the equation converges with order p + 2 when we only use polynomials of degree p + 1 as a finite element space without postprocessing. After establishing the theory, we carry out numerical tests to demonstrate and ensure that the proposed method is effective and accurate in practice.

키워드

과제정보

연구 과제 주관 기관 : Hwa-Rang Dae Research Institute

참고문헌

  1. B.S, Kim and J.T. Oden, hp-version discontinuous Galerkin methods for hyperbolic conservation laws, Computer Methods in Applied Mechanics and Engineering, 133(3) (1996), 259-286. https://doi.org/10.1016/0045-7825(95)00944-2
  2. B. Cockburn and C-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, Journal of Computational Physics, 141(2) (1998), 199-224. https://doi.org/10.1006/jcph.1998.5892
  3. L. Krivodonova, J. Xin, J-F. Remacle, N. Chevaugeon, and J.E. Flaherty, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws, Applied Numerical Mathematics, 48(3) (2004), 323-338. https://doi.org/10.1016/j.apnum.2003.11.002
  4. T.J. Hughes, G. Engel, L. Mazzei, and M.G. Larson, A comparison of discontinuous and continuous Galerkin methods based on error estimates, conservation, robustness and efficiency, Discontinuous Galerkin Methods, (2000), 135-146.
  5. P. Houston and E. Suli, hp-adaptive discontinuous Galerkin finite element methods for first-order hyperbolic problems, SIAM Journal on Scientific Computing, 23(4) (2001), 1226-1252. https://doi.org/10.1137/S1064827500378799
  6. R. Hartmann and P. Houston, Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations, Journal of Computational Physics, 183(2) (2002), 508-532. https://doi.org/10.1006/jcph.2002.7206
  7. R. Biswas, K.D. Devine, and J.E. Flaherty, Parallel, adaptive finite element methods for conservation laws, Applied Numerical Mathematics, 14 (1994), 255-283. https://doi.org/10.1016/0168-9274(94)90029-9
  8. C.E. Baumann and J.T. Oden, A discontinuous hp finite element method for convection-diffusion problems, Computer Methods in Applied Mechanics and Engineering, 175(3) (1999), 311-341. https://doi.org/10.1016/S0045-7825(98)00359-4
  9. P. Castillo, B. Cockburn, D. Schotzau, and C. Schwab, Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems, Mathematics of Computation, 71 (2002), 455-478.
  10. D.N. Arnold, F. Brezzi, B. Cockburn, and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM Journal on Numerical Analysis, 39 (2001), 1749-1779.
  11. B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM Journal on Numerical Analysis, 47 (2009), 1319-1365. https://doi.org/10.1137/070706616
  12. 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
  13. 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
  14. 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
  15. 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
  16. P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, Boston, MA, 1985.
  17. P.G. Ciarlet, The finite element method for elliptic problems, Elsevier North-Holland, New York, 1978.