Optimal transfer operators for nonsymmetric two-grid methods

Abstract

Algebraic Multigrid (AMG) methods have been proven to be effective solvers for large-scale linear algebraic systems $Ax = b$ with Hermitian positive definite (HPD) matrix $A$. For such problems the convergence in the $A$-norm is well understood, but for nonsymmetric indefinite systems fewer results exist. Recently, convergence results for more general $B$-norms induced by certain HPD matrices were established. There, orthogonal projections built by compatible transfer operators are used. Here, we present a theoretical framework for the convergence of nonsymmetric algebraic two-grid methods for arbitrary $B$-inner products and induced $B$-norms which naturally includes the HPD case and all recent results for the nonsymmetric case. For this purpose, we consider two different two-grid error operators with the first one being the natural generalization of the error operator in the HPD case. The second operator has been studied before and is simpler, but requires the additional assumption of normality in some inner product of the smoothing step $M^{-1}A$ to achieve convergence. We prove new convergence results, generalize some previous results and explain the differences and similarities of both operators together with the necessity of the normality. Moreover, we establish optimal compatible interpolation and restriction operators for both two-grid methods that minimize the error norm.