Algebraic multigrid methods based on compatible relaxation and energy minimization

This paper presents an adaptive algebraic multigrid method for the solution of positive definite linear systems arising from the discretizations of elliptic partial differential equations. The proposed method uses compatible relaxation to adaptively construct the set of coarse variables. The nonzero supports for the coarse-space basis is determined by approximation of the so-called two-level “ideal” interpolation operator. Then, an energy minimizing coarse basis is formed using an approach aimed to minimize the trace of the coarse-level operator. The presented approach maintains multigrid-like optimality, without the need for parameter tuning, for some problems where current algorithms exhibit degraded performance. Numerical experiments are presented that demonstrate the efficacy of the approach.