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

TL;DR

This paper investigates spectral properties and develops fast iterative solvers for parameter-dependent curl-div problems in magnetohydrodynamics, using isogeometric analysis and advanced spectral analysis techniques.

Contribution

It provides a detailed spectral analysis of curl-div matrices and introduces efficient multigrid and Krylov solvers tailored for ill-conditioned systems.

Findings

Spectral analysis reveals parameter-dependent behavior of matrices.

Proposed solvers significantly improve computational efficiency.

Numerical tests confirm effectiveness of the methods.

Abstract

Alfv\'en-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 Alfv\'en-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.