Spectral analysis and spectral symbol for the 2D curl-curl (stabilized) operator with applications to the related iterative solutions

Abstract:

In this paper, we study structural and spectral features of linear systems of equations arising from Galerkin approximations of $H(\mathrm {curl})$ elliptic variational problems, based on the Isogeometric Analysis (IgA) approach. Such problems arise in Time Harmonic Maxwell and Magnetostatic problems, as well in the preconditioning of MagnetoHydroDynamics equations, and lead to large linear systems, with different and severe sources of ill-conditioning.

First, we consider a compatible B-splines discretization based on a discrete de Rham sequence and we study the structure of the resulting matrices $\mathcal {A}_{\mathbfit {n}}$. It turns out that $\mathcal {A}_{\mathbfit {n}}$ shows a two-by-two pattern and is a principal submatrix of a two-by-two block matrix, where each block is two-level banded, almost Toeplitz, and where the bandwidths grow linearly with the degree of the B-splines.

Looking at the coefficients in detail and making use of the theory of the Generalized Locally Toeplitz (GLT) sequences, we identify the symbol of each of these blocks, that is, a function describing asymptotically, i.e., for $\mathbfit {n}$ large enough, the spectrum of each block. From this spectral knowledge and thanks to some new spectral tools we retrieve the symbol of $\{\mathcal {A}_{\mathbfit {n}}\}_{\mathbfit {n}}$ which as expected is a two-by-two matrix-valued bivariate trigonometric polynomial. In particular, there is a nice elegant connection with the continuous operator, which has an infinite dimensional kernel, and in fact the symbol is a dyad having one eigenvalue like the one of the IgA Laplacian, and one identically zero eigenvalue; as a consequence, we prove that one half of the spectrum of $\mathcal {A}_{\mathbfit {n}}$, for $\mathbfit {n}$ large enough, is very close to zero and this represents the discrete counterpart of the infinite dimensional kernel of the continuous operator. From the latter information, showing that the considered problem has an ill-posed nature, we are able to give a detailed spectral analysis of the matrices $\mathcal {A}_{\mathbfit {n}}$ and of the corresponding zero-order term stabilized matrices, which is fully confirmed by several numerical evidences.

Finally, by taking into consideration the GLT theory and making use of the spectral results, we furnish indications on the convergence features of known iterative solvers and we suggest a further stabilization technique and proper iterative procedures for the numerical solution of the involved linear systems.