An energy‐based AMG coarsening strategy

Algebraic multigrid (AMG) is an iterative method that is often optimal for solving the matrix equations that arise in a wide variety of applications, including discretized partial differential equations. It automatically constructs a sequence of increasingly smaller matrix problems that hopefully enables efficient resolution of all scales present in the solution. The methodology is based on measuring how a so‐called algebraically smooth error value at one point depends on its value at another. Such a concept of strength of connection is well understood for operators whose principal part is an M‐matrix; however, the strength concept for more general matrices is not yet clearly understood, and this lack of knowledge limits the scope of AMG applicability. The purpose of this paper is to motivate a general definition of strength of connection, discuss its implementation, and present the results of initial numerical experiments.