B-Orthogonal Projections in Nonsymmetric Algebraic Multigrid Methods

Multigrid methods are powerful solvers for large-scale linear systems. Over the past decades, the convergence theory of multigrid methods for symmetric positive definite problems has been thoroughly studied and is currently well understood. However, fewer theoretical results exist for nonsymmetric problems. In [B. S. Southworth and T. A. Manteuffel, SIAM J. Matrix Anal. Appl., 45 (2024), pp. 1245–1258] the concept of so-called compatible transfer operators is introduced for nonsymmetric multigrid methods. Here we use this concept to establish a unified theoretical framework for multigrid methods. This framework contains different existing approaches not only for nonsymmetric methods but also for the symmetric case. We establish sharp convergence estimates, a criterion for optimal transfer (restriction and prolongation) operators is provided, and different convergence measures suitable for particular cases are explored. Moreover, our work generalizes previously well-known results.