Panayot Vassilevski, Lawrence Livermore National Laboratory

KAUST-IAMCS Workshop on Multiscale Modeling, Advanced Discretization Techniques, and Simulation of Wave Propagation

May 7-8, 2011

King Abdullah University of Science and Technology (KAUST)

Thuwal, Kingdom of Saudi Arabia

Finite Element Based Algebraic Multigrid: Principles and Algorithms

The main motivation for using multigrid methods (or MG) for solving systems of (linear) algebraic equations is that they have the potential to be of optimal order. After a brief introduction to MG, we will focus on the algebraic version of the method (or AMG). The latter refers to the case when the hierarchy of vector spaces needed to define a MG algorithm is constructed by the user in a matrix (operator) dependent way. In a sense, constructing an AMG can be viewed as an “inverse” problem and as such it is “ill-posed”; that is, many hierarchies of coarse spaces can be constructed so that they produce equally good (or bad) multigrid methods. We will focus on one class of AMG methods suitable for discretized partial differential equations on unstructured meshes; namely, on the element agglomeration AMG (or AMGe) by pointing out several general approaches to construct the AMG hierarchies, including an adaptive AMG that we will illustrate on the example of Helmholtz equation in a first order system least-squares (or FOSLS) setting.