In this article we present a new formulation for coupling spectral element discretizations to finite difference and finite element discretizations addressing flow problems in very complicated geometries. A general iterative relaxation procedure (Zanolli patching) is employed that enforcesC 1 continuity along the patching interface between the two differently discretized subdomains. In fluid flow simulations of transitional and turbulent flows the high-order discretization (spectral element) is used in the outer part of the domain where the Reynolds number is effectively very high. Near “rough” wall boundaries (where the flow is effectively very viscous) the use of low-order discretizations provides sufficient accuracy and allows for efficient treatment of the complex geometry. An analysis of the patching procedure is presented for elliptic problems, and extensions to incompressible Navier-Stokes equations are implemented using an efficient high-order splitting scheme. Several examples are given for elliptic and flow model problems and performance is measured on both serial and parallel processors.
This is a preview of subscription content,to check access.
Access this article
Patera, A. T. (1984). A spectral element method for fluid dynamics: Laminar flow in a channel expansion.J. Comput. Phys. 54, 468.
Karniadakis, G. E., Bullister, E. T., and Patera, A. T. (1985). A spectral element method for solution of two- and three-dimensional time dependent Navier-Stokes equations. InFinite Element Methods for Nonlinear Problems, Springer-Verlag, New York, p. 803.
Karniadakis, G. E. (1989). Spectral element simulations of laminar and turbulent flows in complex geometries,Applied Numer. Math. 6, 85.
Clouqueur, A., and d'Humières, D. (1987).RAP1, a cellular automaton machine for fluid dynamics,Complex Syst. 1, 585–597.
Nosenchuck, D. M., and Littman, M. G. (1986). The coming age of the parallel processor, In Proc. 23rd Annual Space Conference, NASA.
Karniadakis, G. E., Israeli, M., and Orszag, S. A. (1991). High-order splitting methods for the incompressible Navier-Stokes equations,J. Comput. Phys. 1991, to appear.
Funaro, D., Quarteroni, A., and Zanolli, P. (1985). An interative procedure with interface relaxation for domain decomposition methods. Technical Report 530, Instituto di Analisi Numerica del Consiglio Nazionale delle Ricerche, Pavia.
Maday, Y., and Patera, A. T. (1989). Spectral element methods for the Navier-Stokes equations,State of the Art Surveys in Computational Mechanics, ASME.
Gottlieb, D., and Orszag, S. A. (1977).Numerical Analysis of Spectral Methods: Theory and Applications, SIAM, Philadelphia.
Roache, P. J. (1982).Computational Fluid Dynamics, Hermosa Publishers.
van der Wijngaart, R. G. (1990). Composite-Grid Techniques and Adaptive Mesh Refinement in Comptational Fluid Dynamics, report No. CLaSSiC-90-07, Stanford University, January.
Canuto, C., and Funaro, D. (1987). The Schwarz algorithm for spectral methods,SIAM J. Numer. Anal.
Lions, P. L. (1989). On the Schwarz alternating method III: A variant for non-overlapping subdomains, in Chanet al. (1989), 202.
Douglas, C. C. (1989). A variation of the Schwarz alternating method: The domain decomposition reduction method, in Chanet al. (1989), p. 191.
Kovasznay, L. I. G. (1948). Laminar flow behind a two-dimensional grid, inProc. Cambridge Phil. Society 1948, 44.
Chan, T. F., Glowinski, R., Periaux, J., and Widlund, O. B. (eds.) (1989).Third Int. Symp. on Domain Decomposition Methods for Partial Differential Equations, Houston, Texas, March 1989, SIAM, Philadelphia.
About this article
Cite this article
Henderson, R., Karniadakis, G.E. Hybrid spectral-element-low-order methods for incompressible flows. J Sci Comput 6, 79–100 (1991). https://doi.org/10.1007/BF01062115