Isogeometric analysis for 2D and 3D curl–div problems: Spectral symbols and fast iterative solvers

• Published in: Computer methods in applied mechanics and engineering 2019-02, Vol.344, p.970-997
• Main Authors: Mazza, Mariarosa, Manni, Carla, Ratnani, Ahmed, Serra-Capizzano, Stefano, Speleers, Hendrik
• Format: Article
• Language: English
• Subjects: Alfvén-like operator; Computational fluid dynamics; Dependence; GLT theory; Ill-conditioned problems (mathematics); Isogeometric analysis; Iterative methods; Krylov preconditioning; Linear systems; Magnetohydrodynamics; Multigrid techniques; Numerical methods; Parameters; Plasma physics; Plasmas (physics); Solvers; Spectra; Spectral symbol; Splines; Tensors; Two dimensional analysis
• ISSN: 0045-7825; 1879-2138; 1879-2138
• DOI: 10.1016/j.cma.2018.10.008

Alfvén-like operators are of interest in magnetohydrodynamics, which is used in plasma physics to study the macroscopic behavior of plasma. Motivated by this important and complex application, we focus on a parameter-dependent curl–div problem that can be seen as a prototype of an Alfvén-like operator, and we discretize it using isogeometric analysis based on tensor-product B-splines. The involved coefficient matrices can be very ill-conditioned, so that standard numerical solution methods perform quite poorly here. In order to overcome the difficulties caused by such ill-conditioning, a two-step strategy is proposed. First, we conduct a detailed spectral study of the coefficient matrices, highlighting the critical dependence on the different physical and approximation parameters. Second, we exploit such spectral information to design fast iterative solvers for the corresponding linear systems. For the first goal we apply the theory of (multilevel block) Toeplitz and generalized locally Toeplitz sequences, while for the second we use a combination of multigrid techniques and preconditioned Krylov solvers. Several numerical tests are provided both for the study of the spectral problem and for the solution of the corresponding linear systems. •We focus on the IgA discretization of a parameter-dependent curl–div operator.•The operator is a prototype Alfvén-operator of interest in magnetohydrodynamics.•We provide a symbol-based spectral analysis of the resulting matrices.•We design fast iterative solvers for the obtained linear systems.