Multigrid for Q k Finite Element Matrices Using a (Block) Toeplitz Symbol Approach

Paola Ferrari, Ryma Imene Rahla, Cristina Tablino-Possio, Skander Belhaj and Stefano Serra-Capizzano
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Paola Ferrari: Dipartimento di Scienza ed Alta Tecnologia, Università dell’Insubria-Sede di Como, Via Valleggio 11, 22100 Como, Italy
Ryma Imene Rahla: University of Tunis El Manar, ENIT-LAMSIN, BP 37, Tunis 1002, Tunisia
Cristina Tablino-Possio: Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via Cozzi 53, 20125 Milano, Italy
Skander Belhaj: University of Tunis El Manar, ENIT-LAMSIN, BP 37, Tunis 1002, Tunisia
Stefano Serra-Capizzano: Dipartimento di Scienze Umane e dell’Innovazione per il Territorio, Università dell’Insubria-Sede di Como, Via Valleggio 11, 22100 Como, Italy

Mathematics, 2019, vol. 8, issue 1, 1-17

Abstract: In the present paper, we consider multigrid strategies for the resolution of linear systems arising from the Q k Finite Elements approximation of one- and higher-dimensional elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div − a ( x ) ∇ · , with a continuous and positive over Ω ¯ , Ω being an open and bounded subset of R 2 . While the analysis is performed in one dimension, the numerics are carried out also in higher dimension d ≥ 2 , showing an optimal behavior in terms of the dependency on the matrix size and a substantial robustness with respect to the dimensionality d and to the polynomial degree k .

Keywords: Multigrid; Matrix-sequences; Spectral analysis; Finite Element approximations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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