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

• Published in: Numerical linear algebra with applications Vol. 28; no. 4
• Main Authors: Donatelli, Marco, Ferrari, Paola, Furci, Isabella, Serra‐Capizzano, Stefano, Sesana, Debora
• Format: Journal Article
• Language: English
• Published: Oxford Wiley Subscription Services, Inc 01.08.2021
• Subjects: block‐Toeplitz sequences; Commutativity; Convergence; convergence analysis; Finite element method; Galerkin method; Linear systems; Mathematical analysis; Matrices (mathematics); matrix valued symbol; Multigrid methods; Operators (mathematics); Scalars; spectral analysis
• ISSN: 1070-5325; 1099-1506; 1099-1506
• DOI: 10.1002/nla.2356

In the past decades, multigrid methods for linear systems having multilevel Toeplitz coefficient matrices with scalar entries have been widely studied. On the other hand, only few papers have investigated the case of block entries, where the entries are small generic matrices of fixed size instead of scalars. In that case the efforts of the researchers have been mainly devoted to specific applications, focusing on algorithmic proposals, but with 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 is 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. According to the analysis, we define a class of grid transfer operators satisfying the previous theoretical conditions and we propose a strategy to ensure 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.