Spectral analysis and spectral symbol of matrices in isogeometric collocation methods

Abstract:

We consider a linear full elliptic second order partial differential equation in a $d$-dimensional domain, $d\ge 1$, approximated by isogeometric collocation methods based on uniform (tensor-product) B-splines of degrees $\boldsymbol {p}:=(p_1,\ldots ,p_d)$, $p_j\ge 2$, $j=1,\ldots ,d$. 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 and study 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 non-negative function with a unique zero of order two at $\boldsymbol {\theta }=\mathbf {0}$ (where $\boldsymbol {\theta }:=(\theta _1,\ldots ,\theta _d)$ are the Fourier variables), but with infinitely many ‘numerical zeros’ for large $\|\boldsymbol {p}\|_\infty$. Indeed, the symbol converges exponentially to zero with respect to $p_j$ at all the points $\boldsymbol {\theta }$ such that $\theta _j=\pi$. In other words, if $p_j$ is large, all the points $\boldsymbol {\theta }$ with $\theta _j=\pi$ behave numerically like a zero of the symbol. The presence of the zero of order two at $\boldsymbol {\theta }=\mathbf {0}$ is expected because it is intrinsic in any local approximation method, such as finite differences and finite elements, of second order differential operators. However, the ‘numerical zeros’ lead to the surprising fact that, for large $\|\boldsymbol {p}\|_\infty$, 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 $\boldsymbol {p}$, of standard iterative methods. On the other hand, this 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 $\boldsymbol {p}$.