Structured grid AMG with stencil-collapsing for d-level circulant matrices

SUMMARY We introduce a multigrid method with a stencil collapsing technique for linear systems whose coefficient matrix An ∈ R the eigenvalues of A are given by its generating symbol f , a [0, 2π) d-periodic function, as λ j = f (2πj1/n1, 2πj2/n2,. .. , 2πj d /n d). In the case of banded (multilevel) circulant systems, multigrid is an optimal, i.e. O(n) solver, which is superior to FFT techniques which have complexity O(n log n). As the multigrid technique introduced by Serra et al. is based on the classical AMG theory by Ruge and Stüben, involving the Galerkin product A k = RAnR H for the definition of the coarse grid problem, the technique suffers from stencil growth, a main drawback of algebraic multigrid methods. In the present work, we derive coarse grid operators that are spectrally equivalent to the classical operator defined by the Galerkin operator, while keeping the number of stencil entries constant. To do so, a stencil collapsing technique, which creates sparser stencils, is applied. We will present the technique and show that it produces matrices that are spectrally equivalent to the original matrices for a subclass of d-level circulant matrices.