DOI QR코드

DOI QR Code

LEAST-SQUARES METHOD FOR THE BUBBLE STABILIZATION BY THE GAUSS-NEWTON 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 : 2015.09.21
  • Accepted : 2015.12.22
  • Published : 2016.03.25

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 least-squares solution which minimizes the sum of the squares of errors at each node points. Some numerical results are reported.

Keywords

Least-squares method;Legendre spectral method;bubble-stabilization;advection-diffusion equation;Gauss-Newton method

References

  1. C. BAIOCCHI, F. BREZZI AND L.P. FRANCA Virtual bubbles and Galerkin-least-squares type methods, Comput. Methods. Appl. Mech. Engrg., 105(1993), pp. 125-142. https://doi.org/10.1016/0045-7825(93)90119-I
  2. 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), pp. 199-259. https://doi.org/10.1016/0045-7825(82)90071-8
  3. C. CANUTO Spectral methods and a maximum principle, Math. Comp. 51 (1988) pp615-629. https://doi.org/10.1090/S0025-5718-1988-0930226-2
  4. C. CANUTO, M.Y.HUSSAINI, A. QUARTERONI AND T.A. ZANG Spectral Methods. Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer-Verlag, Berlin 2007
  5. C. CANUTO, M.Y.HUSSAINI, A. QUARTERONI AND T.A. ZANG Spectral Methods. Fundamentals in Single Domains, Springer-Verlag, Berlin 2006
  6. C. CANUTO and G. PUPPO, Bubble stabilization of spectral Legendre methods for the advection-diffusion equation, Comput. Methods Appl. Mech. Engr., 118 (1994), pp. 239-263. https://doi.org/10.1016/0045-7825(94)90002-7
  7. S. D. KIM Piecewise bilinear preconditioning of high-order finite element methods, Electron. Trans. Numer. Anal. 26 (2007), 228-242.
  8. S. KIM and S. D. KIM, Preconditioning on high-order element methods using Chebyshev-Gauss-Lobatto nodes, Applied. Numer. Math., 59 (2009), pp. 316-333. https://doi.org/10.1016/j.apnum.2008.02.007
  9. S. D. KIM and S. PARTER, Preconditioning Chebyshev spectral collocation method for elliptic partial differential equations, SIAM J. Numer. Anal., 33(1996), pp2375-2400. https://doi.org/10.1137/S0036142994275998
  10. SEUNG SOO KIM, YONG HUN LEE AND EUN JUNG OH Optimization for the Bubble Stabilized Legendre Galerkin Method by Steepest Descent Method, Honam Mathematical Journal, 36 (2014), pp755-766. https://doi.org/10.5831/HMJ.2014.36.4.755
  11. H. MA, Chebyshev-Legendre spectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal., 35 (1998), pp869-892. https://doi.org/10.1137/S0036142995293900
  12. W. SUN AND Y. YUAN Optimization Theory and Methods. Nonlinear Programming, Springer-Verlag, 2006