DOI QR코드

DOI QR Code

CONVERGENCE ANALYSIS ON GIBOU-MIN METHOD FOR THE SCALAR FIELD IN HODGE-HELMHOLTZ DECOMPOSITION

  • Min, Chohong (DEPARTMENT OF MATHEMATICS, EWHA WOMANS UNIVERSITY) ;
  • Yoon, Gangjoon (INSTITUTE OF MATHEMATICAL SCIENCES, EWHA WOMANS UNIVERSITY)
  • Received : 2014.10.10
  • Accepted : 2014.11.16
  • Published : 2014.12.25

Abstract

The Hodge-Helmholtz decomposition splits a vector field into the unique sum of a divergence-free vector field (solenoidal part) and a gradient field (irrotational part). In a bounded domain, a boundary condition needs to be supplied to the decomposition. The decomposition with the non-penetration boundary condition is equivalent to solving the Poisson equation with the Neumann boundary condition. The Gibou-Min method is an application of the Poisson solver by Purvis and Burkhalter to the decomposition. Using the $L^2$-orthogonality between the error vector and the consistency, the convergence for approximating the divergence-free vector field was recently proved to be $O(h^{1.5})$ with step size h. In this work, we analyze the convergence of the irrotattional in the decomposition. To the end, we introduce a discrete version of the Poincare inequality, which leads to a proof of the O(h) convergence for the scalar variable of the gradient field in a domain with general intersection property.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. J. B. Bell, P. Colella, and H. M. Glaz, A second order projection method for the incompressible Navier-Stokes equations, J. Comput. Phys., 85 (1989), 257-283. https://doi.org/10.1016/0021-9991(89)90151-4
  2. S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 2008.
  3. A. Chorin, A numerical method for solving incompressible ciscous flow problems, J. Comput. Phys., 2 (1967), 12-26. https://doi.org/10.1016/0021-9991(67)90037-X
  4. F. Gibou and C. Min, Efficient symmetric positive definite second-order accurate monolithic solver for fluid/solid interactions, J. of Comput. Phys., 231 (2012), 3246-3263. https://doi.org/10.1016/j.jcp.2012.01.009
  5. F. Harlow and J. Welch, Numerical calculation of time-dependent viscous incompressible flow of fluids with free surfaces, Physics of Fluids, 8 (1965), 2182-2189. https://doi.org/10.1063/1.1761178
  6. H. Helmholtz, On integrals of the hydrodynamic equations which correspond to vortex motions, Journal fur die reine und angewandte Mathematik, 55 (1858), 22-55.
  7. J. Kim and P. Moin, Application of a Fractional-step Method to Incompressible Navier-Stokes Equations, J. Comput. Phys., 59 (1985), 308-323. https://doi.org/10.1016/0021-9991(85)90148-2
  8. W. E. and J. G. Liu, Gauge method for viscous incompressible flows, Comm. Math. Sci., 1 (2003), 317-332. https://doi.org/10.4310/CMS.2003.v1.n2.a6
  9. Y.-T. Ng, C. Min and F. Gibou, An efficient fluid-solid coupling algorithm for single-phase flows, J. Comput. Phys., 228 (2009), 8807-8829. https://doi.org/10.1016/j.jcp.2009.08.032
  10. C. Pozrikidis, Introduction to theoretical and computational fluid dynamics, Oxford university press, 1997.
  11. J. W. Purvis and J. E. Burhalter, Prediction of critical Mach number for store configurations, AIAA J., 17 (1979), 1170-1177. https://doi.org/10.2514/3.7617
  12. G. Yoon and C. Min, On treating grid nodes too near the boundary in Shortley-Weller method J. Comput. Phys., (2014), submitted.
  13. G. Yoon, J. Park, and C. Min, Convergence analysis on Gibou-Min method for the Hodge projection, Communications in Mathematical Sciences, (2014), submitted.