A Symbol-Based Analysis for Multigrid Methods for Block-Circulant and Block-Toeplitz Systems

In the literature, there exist several studies on symbol-based multigrid methods for solution of linear systems having structured coefficient matrices. particular, convergence analysis such has been obtained in an elegant form case Toeplitz matrices generated by a scalar-valued function. block-Toeplitz setting, that is, where matrix entries are small generic instead scalars, some algorithms have already proposed regarding specific applications, and first rigorous performed [M. Donatelli et al., Numer. Linear Algebra Appl., 28 (2021), e2356]. However, with existent theoretical tools, it is still not possible to prove many known literature. This paper aims generalize previous results, giving more general sufficient conditions symbol grid transfer operators. we treat matrix-valued trigonometric polynomials which can be nondiagonalizable singular at all points, express new terms eigenvectors associated ill-conditioned subspace. Moreover, extend V-cycle method, proving rate under stronger conditions, resemble those given scalar case. order validate our findings, present classical block problem stemming from FEM approximation second differential problem. We focus two strategies use geometric standard bisection operators both fall into category projectors satisfying conditions. addition, using tensor product argument, provide strategy construct efficient procedures multilevel setting.