Multigrid methods for block-Toeplitz linear systems: convergence analysis and applications

TL;DR

This paper develops a general convergence analysis for multigrid methods applied to positive definite block Toeplitz systems, providing theoretical guarantees and practical strategies for efficient solutions in complex applications.

Contribution

It introduces a novel two-grid convergence proof for block Toeplitz matrices with generic blocks, including a new approximation property and transfer operator design.

Findings

Optimal convergence rate independent of matrix size

Effective multigrid strategies for high-order finite element methods

Numerical results confirm theoretical predictions and efficiency

Abstract

In the past decades, multigrid methods for linear systems having multilevel Toeplitz coefficient matrices with scalar entries have been largely studied. On the other hand, only few papers have investigated the case of block entries, where the entries are small generic matrices instead of scalars. In that case the efforts of the researchers have been mainly devoted to specific applications, focusing on algorithmic proposals but with very marginal theoretical results. In this paper, we propose a general two-grid convergence analysis proving an optimal convergence rate independent of the matrix size, in the case of positive definite block Toeplitz matrices with generic blocks. In particular, the proof of the approximation property has not a straightforward generalization of the scalar case and in fact we have to require a specific commutativity condition on the block symbol of the grid transfer operator. Furthermore, we define a class of grid transfer operators satisfying the previous theoretical conditions and we propose a strategy to insure fast multigrid convergence even for more than two grids. Among the numerous applications that lead to the block Toeplitz structure, high order Lagrangian finite element methods and staggered discontinuous Galerkin methods are considered in the numerical results, confirming the effectiveness of our proposal and the correctness of the proposed theoretical analysis.