In this investigation we examine the use of one-way cascadic multigrid strategies CMG for solution of incompressible viscous flow problems using the finite element method. The content of the presentation is as follows: First we describe the basic CMG approach for representative elliptic boundary value problems and summarize the theoretical error estimates from approximation theory, desired smoother properties, and arithmetic complexity of the method as a function of iteration parameters at each level. The extension of these error and complexity estimates to adaptive grids is also given.
In the numerical experiments, performance of the algorithm on both serial and distributed parallel systems is examined. We carry out a series of comparison studies between the CMG scheme and a standard bi-conjugate gradient solve strategy on the fine level grid. Other issues such as the treatment of the diagonal elements in the zero block corresponding to the pressure variables are also considered and the results of associated numerical studies are presented. Finally, parallel performance studies on a distributed parallel PC cluster in the CFDLab are described and both performance and flow simulation results are given for 3D applications to linear Stokes flow, low Reynolds number Navier Stokes problems, and coupled fluid-thermal problems.
Bornemann, F. and P. Deuflhard, The Cascadic Multigrid Method for Elliptic Problems, Num. Math, 75, 135-152, 1996.
Bramble, J., J. Pasciak, J. Wang and J. Xu, Convergence Estimates for Multigrid Algorithms without Regularity Assumptions, Math. Comp, 57, 23-45, 1991.
Carey, G. F., Computational Grids: Generation, Adaptation and Solution Strategies, Taylor & Francis, 1997.
Carey, G. F., Adaptive Techniques and Related Issues in Finite Element Modeling of Heat and Fluid Flow, To appear in Proceedings of CHT'01 (Australia May 2001).
Carey, G. F., R. McLay, G. Bicken, B. Barth, S. Swift and A. Ardelea, Parallel Finite Element Solution of 3D Rayleigh-Benard-Marangoni Flows, IJNMF, 31, 37-52, 1999.
Carey, G. F., R. McLay, W. Barth, S. Swift and B. Kirk, Distributed Parallel Simulation of Surface Tension Driven Viscous Flow and Transport Processes, Submitted to World Scientific, June, 2000.
Ciarlet, P. J., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
Deuflhard, P., Cascadic Conjugate Gradient Methods for Elliptic Partial Differential Equations: Algorithm and Numerical Results, in D. Keyes and J. Xu (eds)., Domain Decomposition Methods in Scientific and Engineering Computing, AMS Series, 180, 29-42, 1994.