Spectral Analysis of Isogeometric Discretizations of 2D Curl-Div Problems with General Geometry

We study the spectral distribution of matrices arising in Galerkin isogeometric methods for weighted curl-div operators defined on a general planar domain. It can be compactly described by means of a spectral symbol which depends on the characteristic parameters of the problem: the weight parameters, the basic curl and div operators, the degree of the B-spline approximation, and the geometry map used to represent the computational domain.

Dirichlet (no-slip) boundary conditions, i.e., u = 0 on ∂ . This leads us to the following variational formulation We refer the reader to [3,15] for a discussion about well-posedness.
To find an approximate solution of the problem L α,β u = f , with the stated boundary conditions, we consider the variational formulation (2) in a finite dimensional vector space V h ⊂ H 1 0 ( ) 2 , i.e., We focus on isogeometric analysis (IgA) as discretization technique, where the approximation space V h is chosen to be composed of vector fields whose components are linear combinations of tensor-product B-splines mapped according to G. The discretization (3) leads to solving linear systems, which turn out to be severely ill-conditioned and require ad hoc fast solvers for a proper treatment [4,6,15]. This requires a deep understanding of the spectral properties of the related matrices. They depend on many factors: the problem parameters α, β, the basic curl and div operators, the mesh-size, the degree of the B-spline approximation, and the map G used to describe the geometry of the computational domain.
In this paper we provide a spectral study of these matrices using the theory of (multilevel block) Toeplitz [13,17,19] and generalized locally Toeplitz [10][11][12] sequences. More precisely, we show that such matrices admit a spectral distribution which can be described in terms of a so-called spectral symbol. We determine this spectral symbol and we reveal its dependence on the characteristic parameters of the problem listed above. The spectral analysis presented in this paper extends the results of [15] to the case of non-trivial geometry and relies on the spectral theory developed for isogeometric discretizations of elliptic problems in [7,8]. We also refer the reader to [16] for a spectral analysis of the curl-curl operator.
The remainder of the paper is organized as follows. In Sect. 2 we introduce notations and definitions relevant for our spectral analysis, and we recall the basics of B-splines. In Sect. 3 we detail the IgA discretization matrices and we perform a spectral analysis of them. We numerically illustrate those results in Sect. 4. Finally, we conclude the paper in Sect. 5.

Preliminaries
In this section we collect some preliminary tools on spectral analysis and IgA discretizations. In particular, we recall the formal definition of spectral distribution for a general matrix-sequence and the definition of (cardinal) B-splines.

Spectral Distribution
Throughout the paper, we follow the standard convention for operations with multiindices (see e.g. [9,18]). Given a multi-index n := (n 1 , . . . , n d ) ∈ N d , we say n → ∞ if n i → ∞, i = 1, . . . , d. Let C 0 (C) be the set of continuous functions F : C → C with compact support.
Definition 1 Let f : D → C s×s be a measurable matrix-valued function, defined on a measurable set D ⊂ R q with q ≥ 1, 0 < μ q (D) < ∞, where μ q is the Lebesgue measure. Let {A n } n be a matrix-sequence with dim(A n ) =: d n and d n → ∞ as n → ∞. Then, {A n } n is distributed like the pair (f, D) in the sense of the eigenvalues, denoted by {A n } n ∼ λ (f, D), if the following limit relation holds for all F ∈ C 0 (C): where λ j (A n ), j = 1, . . . , d n are the eigenvalues of A n and λ i (f ), i = 1, . . . , s are the eigenvalues of f . We say that f is the (spectral) symbol of the matrix-sequence {A n } n .
If f is smooth enough and the matrix-size of A n is sufficiently large, then the limit relation (4) has the following informal meaning: a first set of d n /s eigenvalues of A n is approximated by a sampling of λ 1 (f ) on a uniform equispaced grid of the domain D, a second set of d n /s eigenvalues of A n is approximated by a sampling of λ 2 (f ) on a uniform equispaced grid of the domain D, and so on, up to few outliers.
In general, understanding whether a matrix-sequence admits a symbol and how to compute it is not an easy task. On the other hand, any "reasonable" approximation of partial differential equations by local methods leads to matrix-sequences that are in the so-called generalized locally Toeplitz (GLT) algebra, and so admit a symbol [10][11][12]. The IgA discretization of our curl-div problem (3) fits in this frame.

B-Splines
For p ≥ 0 and n ≥ 1, consider the uniform knot sequence This knot sequence allows us to define n + p B-splines of degree p. Let χ I denote the characteristic function on the interval I .
where a fraction with zero denominator is assumed to be zero.
It is well known (see e.g. [1]) that the B-splines N p i , i = 1, . . . , n + p, form a basis, and The central B-splines N p i , i = p + 1, . . . , n, are uniformly shifted and scaled versions of a single shape function, the so-called cardinal B-spline φ p : R → R, More precisely, we have The cardinal B-spline φ p is a C p−1 function which is locally supported on the interval [0, p + 1]. Finally, we recall the definition of tensor-product B-splines.

Definition 3
The tensor-product B-splines of bi-degree p := (p 1 , p 2 ) over a uniform mesh of [0, 1] 2 , consisting of n := (n 1 , n 2 ) intervals in each direction, are denoted by N p i : [0, 1] 2 → R, i = 1, . . . , n + p, and defined as We define the tensor-product spline space S Note that all the elements of this space vanish at the boundary of [0, 1] 2 ; see (5). Hence, the space incorporates homogeneous Dirichlet boundary conditions.

Spectral Analysis of Isogeometric Discretizations in 2D
Suppose that the physical domain can be described by a global geometry map, G := [G 1 , G 2 ] T , G : → , which is invertible in the parametric domain := [0, 1] 2 and satisfies G(∂ ) = ∂ . Let where φ p,1 and for k l ∈ {i l , j l }, l = 1, 2, Then, we setφ k 1 ,k 2 = N p k 1 ⊗ N p k 2 , i.e., the tensor-product B-splines in (6). For simplicity of notation, we have taken n 1 = n 2 = n and p 1 = p 2 = p. Also note that In the following, we start by discussing the coefficient matrices arising from the IgA discretization of a generalized Poisson problem. Then, we construct the coefficient matrices related to the IgA discretization of our curl-div problem (3) using (7), and we perform a spectral analysis.

Matrices Related to a Generalized Poisson Problem
Let us focus on the following bivariate generalized Poisson operator: where K : → R 2×2 , and consider homogeneous Dirichlet boundary conditions, i.e., u = 0 on ∂ . From [8] we know that the Galerkin discretization of (8) using one component of the space (7) leads to the coefficient matrix A p,K n,G defined by It has been proved in [8] that such matrices admit a spectral distribution according to Definition 1. To this end, let us define wherex ∈ [0, 1] 2 , θ ∈ [−π, π] 2 , and • is the Hadamard matrix product.
We refer the reader to [8,9] for a detailed discussion about the symbol (9).

Matrices Related to Our Curl-Div Problem
We can reformulate (1) in 2D as where u(x 1 , When discretizing the weak form (3) using the space (7)  .
The blocks related to the curl-curl operator (∇ × ·, ∇ × ·) are given by 12 . Note that all those blocks are symmetric matrices. Similarly, the blocks related to the div-div operator (∇·, ∇·) are given by (see also (10) In the next subsection we compute the symbol of the matrix-sequence {A p,α,β n,G } n .
In Fig. 1 we numerically check relation (11) by comparing the eigenvalues of A p,α,β n,G with the values collected in = { 1 , 2 }, ordered in ascending way, for α = 1 and β = 0.1. We observe that, in a complete agreement with the theory, the considered sampling of λ i (f p,α,β G ), i = 1, 2, describes quite accurately the behavior of the eigenvalues of A p,α,β n,G , also for relatively small matrix-sizes, up to few outliers.

Conclusions
We have analyzed the spectral properties of matrix-sequences arising from isogeometric Galerkin methods for weighted curl-div operators on general planar domains, considering a non-trivial geometry map. More precisely, we have shown that an (asymptotic) spectral distribution exists and it is compactly described by a 2 × 2 spectral symbol. In other words, the eigenvalues of the matrices we are dealing with can be approximated accurately by a uniform sampling of the two eigenvalue functions of the 2 × 2 symbol matrix. The symbol depends on the characteristic parameters of the problem and on the geometry of the physical domain. Its formal structure nicely mimics the structure of the differential problem. The numerical results show a very good matching between the true eigenvalues and the estimates provided by the symbol, already for relatively small matrix-sizes.
The convergence of iterative solvers for linear systems strongly depends on the spectral behavior of the corresponding coefficient matrices. Since the symbol gives a precise description of the spectrum of the curl-div matrix A p,α,β n,G , it could be helpful in the design of good preconditioners that lead to better performance than current solution strategies, like the one in [15,Sect. 5].
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.