Flexible Multiple Semicoarsening for Three-Dimensional Singularly Perturbed Problems

Abstract

Abstract. We present robust parallel multigrid-based solvers for 3D scalar partial differential equations. The robustness is obtained by combining multiple semicoarsening strategies, matrixdependent transfer operators, and a Krylov subspace acceleration. The basis for the 3D preconditioner is a 2D method with multiple semicoarsened grids based on the MG-S method from [C. W. Oosterlee, Appl. Numer. Math., 19(1995) , 3(1991), pp. 1-75] and is evaluated as a solver and as a preconditioner on a MIMD machine. The robustness of the 3D method is shown for finite volume and finite difference discretizations of 3D anisotropic diffusion equations and convection-dominated convection-diffusion problems.

A parallel multigrid-based preconditioner for the 3D heterogeneous high-frequency Helmholtz equation

“…The 2D point-wise Jacobi smoother is generalized here to a line-wise Jacobi smoother, updating all points in the uncoarsened, x 3 -direction simultaneously. The robustness and efficiency of the present 3D multigrid method should be at least similar to a 3D multigrid method based on standard 3D grid coarsening and point-wise smoothing [37].…”

“…4. A particular robust variant of multigrid with 2D semi-coarsening has been introduced in [37] with corresponding analysis. This approach is based on 2D problems [12] where the 2D standard multigrid is employed.…”

“…The interpolation operator can be based on bilinear interpolation or a matrix-dependent interpolation. We choose for the matrix-dependent interpolation as in [37,38], which is especially useful for problems with strong heterogeneities. For the complex Helmholtz preconditioner, we modify the weights in the prolongation operator.…”

“…On the coarse grids, due to the lumping, asymmetry enters the definition of the matrices. Therefore, we adopt the asymmetric strategy as presented in [37] …”

Multiple semi-coarsened multigrid method with application to large eddy simulation

“…While an optimal parallel implementation of the present MSG multigrid method was not the focus of this research, this section brie y describes the approach taken and gives some speedup results. For more detailed investigations of parallel implementations of MSG multigrid, the reader is directed to [26,27].…”