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

SUMMARYWe introduce amultigrid method with a stencil collapsing techniquefor linear systemswhose coefficientmatrix A n ∈ R n× , n = n 1 n 2 ···n d is d-level circulant, that is, independent of n 1 ,n 2 ,...,n d ,the eigenvalues of A are given by its generating symbol f, a [0,2π) d -periodic function, as λ j =f(2πj 1 /n 1 ,2πj 2 /n 2 ,...,2πj d /n d ). In the case of banded (multilevel) circulant systems, multigrid is anoptimal, i.e. O(n) solver, which is superior to FFT techniques which have complexity O(nlogn). Asthe multigrid technique introduced by Serra et al. is based on the classical AMG theory by Ruge andStu¨ben, involving the Galerkin product A k = RA n R H for the definition of the coarse grid problem,the technique suffers from stencil growth, a main drawback of algebraic multigrid methods. In thepresent work, we derive coarse grid operators that are spectrally equivalent to the classical operatordefined by the Galerkin operator, while keeping the number of stencil entries constant. To do so, astencil collapsing technique, which creates sparser stencils, is applied. We will present the techniqueand show that it produces matrices that are spectrally equivalent to the original matrices for a subclassof d-level circulant matrices. Copyright c 2007 John Wiley & Sons, Ltd.key words: circulant matrices; multilevel matrices; two-grid and multigrid iterations