Generalized locally Toeplitz sequences and some applications to MHD subproblems

Day - Time: 30 July 2018, h.11:00
Place: Area della Ricerca CNR di Pisa - Room: C-29
  • Mariarosa Mazza (Max Planck Institute for Plasma Physics, Garching bei Munchen, Germany)

Leonardo Robol


When discretizing a linear PDE by a linear numerical method, the computation of the numerical solution reduces to solving a linear system. The size of this system grows when the discretization parameter n increases, i.e., when we the discretization mesh. We are then in the presence of a sequence of linear systems fAnxn = bng with increasing size.
It is usually observed in practice that the corresponding sequence of discretization matrices fAng inherits a structure from the continuous problem and enjoys an asymptotic spectral distribution compactly described by a function known as symbol. The knowledge of the symbol and of its properties has a very crucial role, since it can be conveniently used for designing fast solvers (of preconditioned Krylov or multigrid type) and/or for analyzing their convergence properties.
The main tool for computing the symbol of a PDE discretization matrix is the theory of Generalized Locally Toeplitz (GLT) matrix-sequences, a .-algebra of matrix-sequences that virtually contains any sequence of matrices coming from `reasonable' approximations by local discretization methods of PDEs (see [1]).
In this talk, we give an overview of the GLT theory and we outline its application to the MagnetoHydro- Dynamics (MHD) equations, which are used in plasma physics to study the macroscopic behavior of plasma. In more detail, we focus on the highly ill-conditioned linear systems coming from a B-Spline discretization of certain parameter-dependent MHD subproblems, and we use the related symbol to design an iterative strategy able to satisfactory deal with the sources of ill-conditioning (see [2, 3]).

[1] C. Garoni and S. Serra-Capizzano, Generalized locally Toeplitz Sequences: Theory and Applications. Vol. I, Springer, Cham, 2017.
[2] M. Mazza, C. Manni, A. Ratnani, S. Serra-Capizzano, and H. Speleers, Isogeometric analysis for 2D and 3D curl-div problems: Spectral symbols and fast iterative solvers, ArXiv:1805.10058, 2018.
[3] M. Mazza, A. Ratnani, and S. Serra-Capizzano, Spectral analysis and spectral symbol for the 2D curl-curl (stabilized) operator with applications to the related iterative solutions, Mathematics of Computation (2018), in press.