In this study, we describe an Algebraic Multigrid Method (AMG) which is a modification of F. Kickinger's graph-based AMG [1]. In Kickinger's AMG, coarsening is based on the graph of the system matrix only, instead of strongly connected components in the matrix, as it is in the "classical" AMG by Ruge and Stüben [2]. This leads to fast computation of both coarse grid matrices and interpolation operators between coarse and fine levels. By some minor changes in the selection of coarse grid variables and processing of Dirichlet boundary conditions, we have obtained more robustness to the method. Additionally, we have extended the method in a way presented e.g. in [3] to be more suitable for vector-valued problems.

In numerical tests, we have used our method as well as a solver as a preconditioner for linear systems of equations arising from discretization of various mathematical models of physical phenomena, such as fluid flow, acoustics, linear elasticity, etc. Here we will show numerical results for two different application areas:

1) incompressible, viscous flow problems in 3-D, and,

2) scattering of sound waves in 2-D.

In the first case, AMG is used as a solver for the Oseen problem arising from Picard-type linearization of steady state Navier-Stokes. Linear Lagrangian finite elements are used as a discretization, stabilized by the well-known scheme by Franca, Frey and Hughes. Incomplete LU factorization with relaxation in smoothing steps is employed as a smoother. In the second case, our method is used to approximate inverse of the shifted-Laplacian operator which is used as a preconditioner for the Helmholz equation. In [4], this was done by using a geometric multigrid. The equations are discretized by linear, quadratic, and cubic Lagrangian finite elements. For both cases we have obtained good convergence results, but the stabilization scheme in the first case restricts the number of coarse levels.

References

[1] F. Kickinger, Algebraic Multi-grid Solver for Discrete Elliptic Second-Order Problems, in: Multigrid Methods V (Stuttgart, 1996), Springer, Berlin, 1998, 157-172.
[2] J.W. Ruge and K. Stüben, Algebraic Multigrid (AMG), in: S.F. McCormick (Ed.), Multigrid Methods, Frontiers in Applied Mathematics, SIAM, Philadelphia, Pennsylvania, 1987, pp. 73-130.
[3] M. Wabro, Algebraic Multigrid Methods for the Numerical Solution of the Incompressible Navier-Stokes Equations, Ph.D. thesis, Johannes Kepler Universität, Linz, 2003.
[4] Y. A. Erlangga, C. W. Oosterlee, C. Vuik, A novel multigrid based preconditioner for heterogeneous Helmholtz problems, SIAM J. Sci. Comput. 27 (4) (2006) 1471 1492.