Spectral analysis of matrices in isogeometric collocation methods

We consider a linear full elliptic second order partial differential equation in a d-dimensional domain, d ≥ 1, approximated by isogeometric collocation methods based on uniform B-splines of degrees p := (p1, . . . , pd), pj ≥ 2, j = 1, . . . , d. We give a construction of the inherently non-symmetric matrices arising from this approximation technique and we perform an analysis of their spectral properties. In particular, we find the associated (spectral) symbol, that is the function describing their asymptotic spectral distribution (in the Weyl sense), when the matrix-size tends to infinity or, equivalently, the fineness parameters tend to zero. The symbol is a nonnegative function with a unique zero of order two at θ = 0 (with θ the Fourier variables), but with infinitely many numerical zeros for large ‖p‖∞, showing up at θj = π if pj is large. The presence of a zero of order two at θ = 0 is expected, because it is intrinsic in any local approximation method of differential operators, like finite differences and finite elements. However, the second type of zeros leads to the surprising fact that, for large ‖p‖∞, there is a subspace of high frequencies where the collocation matrices are ill-conditioned. This non-canonical feature is responsible for the slowdown, with respect to p, of standard iterative methods. On the other hand, its knowledge and the knowledge of other properties of the symbol can be exploited to construct iterative solvers with convergence properties, independent of the fineness parameters and of the degrees p.