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Team for Advanced Flow Simulation and Modeling



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Multiscale Successive Update Method

The Enhanced-Discretization Successive Update Method (EDSUM) [1-4] was proposed as a multiscale iteration method for computation of the flow behavior at small scales. It has a built-in mechanism for transferring flow information between the large and small scales. This information transfer is consistent with the discretizations resulting from the underlying stabilized formulations. This is accomplished without assuming that the small-scale trial or test functions vanish at the borders between the neighboring large-scale elements of the enhanced-discretization zones. The small-scale flow patterns can move from one large-scale element to another without any constraints at the border between the two elements. For iterative solution of the coupled equations systems, the multiscale function space concept is exploited for designing preconditioners based on successive updates corresponding to the large and small scales (see [1-4]). Test computations with the EDSUM were reported in [5].

Fig. 1. Solutions obtained for an advection-dominated problem with the multiscale (top) and standard (bottom) finite element discretizations. The pictures show, from left to right, the solutions obtained with diagonal preconditions after 3, 6, and 9 outer GMRES iterations. For more details, see [5].

Fig. 2. Convergence performance of the multiscale (EDSUM) preconditioners that are based on successive updates corresponding to the large and small scales. Compared to the diagonal preconditioners, the multiscale preconditioners exhibit dramatic increases in convergence speeds. For more details, see [5].

References

1. T.E. Tezduyar, "Finite Element Methods for Flow Problems with Moving Boundaries and Interfaces", Archives of Computational Methods in Engineering, 8 (2001) 83-130.

2. T.E. Tezduyar, "Stabilized Finite Element Methods for Flows with Moving Boundaries and Interfaces", HERMIS: The International Journal of Computer Mathematics and its Applications, 4 (2003) 63-88

3. T.E. Tezduyar, "Finite Element Methods for Fluid Dynamics with Moving Boundaries and Interfaces", Chapter 17 in Encyclopedia of Computational Mechanics, Volume 3: Fluids (eds. E. Stein, R. De Borst and T.J.R. Hughes), John Wiley & Sons (2004).

4. T.E. Tezduyar, "Moving Boundaries and Interfaces", Finite Element Methods: 1970's and Beyond (eds. L.P. Franca, T.E. Tezduyar and A. Masud), CIMNE, Barcelona (2004) 205-220.

5. T.E. Tezduyar and S. Sathe, "Enhanced-Discretization Successive Update Method (EDSUM)", International Journal for Numerical Methods in Fluids, 47 (2005) 633-654.