A Note on Multigrid Methods for (Multilevel)Structured-plus-banded Uniformly BoundedHermitian Positive Definite Linear Systems

In the past few years a lot of attention has been paid in the multigrid solution of multilevel structured (Toeplitz, circulants, Hartley, sine (τ class) and cosine algebras) linear systems, in which the coefficient matrix is banded in a multilevel sense and Hermitian positive definite. In the present paper we provide some theoretical results on the optimality of an existing multigrid procedure, when applied to a properly related algebraic problem. In particular, we propose a modification of previously devised multigrid procedures in order to handle Hermitian positive definite structured-plus-banded uniformly bounded linear systems, arising when an indefinite, and not necessarily structured, banded part is added to the original coefficient matrix. In this context we prove the Two-Grid method optimality. 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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