Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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OPTIMIZATION FOR THE BUBBLE STABILIZED LEGENDRE GALERKIN METHODS BY STEEPEST DESCENT METHOD

  • Kim, Seung Soo (Department of Mathematics (Institute of Pure and Applied Mathematics), Chonbuk National University) ;
  • Lee, Yong Hun (Department of Mathematics (Institute of Pure and Applied Mathematics), Chonbuk National University) ;
  • Oh, Eun Jung (Department of Mathematics (Institute of Pure and Applied Mathematics), Chonbuk National University)
  • Received : 2014.08.12
  • Accepted : 2014.09.29
  • Published : 2014.12.25
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Abstract

In the discrete formulation of the bubble stabilized Legendre Galerkin methods, the system of equations includes the artificial viscosity term as the parameter. We investigate the estimation of this parameter to get the optimal solution which minimizes the maximum error. Some numerical results are reported.

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References

  1. A.N.T. Brooks and T.J.R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convected dominated flows with a particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982), 199-259. https://doi.org/10.1016/0045-7825(82)90071-8
  2. C. Baiocchi, F. Brezzi and L.P. Franca, Virtual bubbles and Galerkin-leastsquares type methods, Comput. Methods. Appl. Mech. Engrg. 105 (1993), 125-142. https://doi.org/10.1016/0045-7825(93)90119-I
  3. F. Brezzi, M.-O. Bristeau, L.P. Franca, M. Mallet and G. Roge, A relationship between stabilized finite element methods and the Galerkin Method with bubble functions, Comput. Methods. Appl. Mech. Engrg. 96 (1992), 117-130. https://doi.org/10.1016/0045-7825(92)90102-P
  4. C. Canuto, Spectral methods and a maximum principle, Math. Comp. 51 (1988), 615-629. https://doi.org/10.1090/S0025-5718-1988-0930226-2
  5. C. Canuto, Stabilization of spectral methods by finite element bubble functions, in: C. Beranrdi and Y. Maday, eds., Proc. ICOSAHOM '92 Conf., Montpellier, 1992 (North-Holland, Amsterdam, 1994); also: Comput. Methods Appl. Mech. Engrg. 116 (1994), 13-26.
  6. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods. Evolution to Complex Geometries and Applications to Flid Dynamics, Springer-Verlag, Berlin, 2007.
  7. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods. Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.
  8. C. Canuto and G. Puppo, Bubble stabilization of spectral Legendre methods for the advection-diffusion equation, Comput. Methods Appl. Mech. Engr. 118 (1994), 239-263. https://doi.org/10.1016/0045-7825(94)90002-7
  9. S.D. Kim, Piecewise bilinear preconditioning of high-order finite element methods, Electron. Trans. Numer. Anal. 26 (2007), 228-242.
  10. S. Kim and S.D. Kim, Preconditioning on high-order element methods using Chebyshev-Gauss-Lobatto nodes, Applied. Numer. Math. 59 (2009), 316-333. https://doi.org/10.1016/j.apnum.2008.02.007
  11. S.D. Kim and S. Parter, Preconditioning Chebyshev spectral collocation method for elliptic partial differential equations, SIAM J. Numer. Anal. 33 (1996), 2375-2400. https://doi.org/10.1137/S0036142994275998
  12. J.H. Lee, Bubble Stabilization of Chebyshev Spectral Method for Advection-Diffusion Equation, Ph.D. Thesis, KAIST, Korea (1988).
  13. H. Ma, Chebyshev-Legendre spectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal. 35 (1998), 869-892. https://doi.org/10.1137/S0036142995293900
  14. W. Sun and Y. Yuan, Optimization Theory and Methods. Nonlinear Programming, Springer-Verlag, 2006.

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