Algebraic multigrid and 4th-order discrete-difference equations of incompressible fluid flow

This paper investigates the effectiveness of two different Algebraic Multigrid (AMG) approaches to the solution of 4th-order discrete-difference equations for incompressible fluid flow (in this case for a discrete, scalar, stream-function field). One is based on a classical, algebraic multigrid, method (C-AMG) the other is based on a smoothed-aggregation method for 4th-order problems (SA-AMG). In the C-AMG case, the inter-grid transfer operators are enhanced using Jacobi relaxation. In the SA-AMG case, they are improved using a constrained energy optimization of the coarse-grid basis functions. Both approaches are shown to be effective for discretizations based on uniform, structured and unstructured, meshes. They both give good convergence factors that are largely independent of the mesh size/bandwidth. The SA-AMG approach, however, is more costly both in storage and operations. The Jacobi-relaxed C-AMG approach is faster, by a factor of between 2 and 4 for two-dimensional problems, even though its reduction factors are inferior to those of SA-AMG.

For non-uniform meshes, the accuracy of this particular discretization degrades from 2nd to 1st order and the convergence factors for both methods then become mesh dependent. Copyright © 2009 John Wiley & Sons, Ltd.