Two-Grid Methods for Hermitian positive definite linear systems connected with an order relation

Given a multigrid procedure for linear systems with coefficient matrices $$A_n,$$ we discuss the optimality of a related multigrid procedure with the same smoother and the same projector, when applied to properly related algebraic problems with coefficient matrices $$B_n$$: we assume that both $$A_n$$ and $$B_n$$ are Hermitian positive definite with $$A_n\le \vartheta B_n,$$ for some positive $$\vartheta $$ independent of $$n.$$ In this context we prove the Two-Grid Method optimality. We apply this elementary strategy for designing a multigrid solution for modifications of multilevel structured linear systems, in which the Hermitian positive definite coefficient matrix is banded in a multilevel sense. As structured matrices, Toeplitz, circulants, Hartley, sine ($$\tau $$ class) and cosine algebras are considered. In such a way, several linear systems arising from the approximation of integro–differential equations with various boundary conditions can be efficiently solved in linear time (with respect to the size of the algebraic problem). Some numerical experiments are presented and discussed, both with respect to Two-Grid and multigrid procedures.

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