Isogeometric analysis for 2D and 3D curl-div problems: Spectral symbols and fast iterative solvers
For researchers in computational plasma physics and isogeometric analysis, this work provides a systematic approach to solve a challenging class of problems, though it is incremental in combining existing techniques.
The paper addresses ill-conditioned linear systems arising from isogeometric discretizations of parameter-dependent curl-div problems (Alfvén-like operators). It proposes a two-step strategy: spectral analysis using Toeplitz theory to understand parameter dependence, and fast iterative solvers combining multigrid and preconditioned Krylov methods, with numerical tests demonstrating effectiveness.
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.