On the matrices in B-spline collocation methods for Riesz fractional equations and their spectral properties

TL;DR

This paper analyzes the spectral properties of matrices arising from B-spline collocation methods for Riesz fractional equations, revealing their structure, conditioning issues, and approximation behavior for different polynomial degrees.

Contribution

It characterizes the spectral properties of the coefficient matrices, showing their Toeplitz-like structure and conditioning behavior, and derives approximation orders for fractional B-spline collocation.

Findings

Matrices are Toeplitz-like with a symbol having a zero at zero of order α.

Conditioning deteriorates at high frequencies as polynomial degree increases.

Approximation order depends on polynomial degree and fractional order, being p+2-α for even p and p+1-α for odd p.

Abstract

In this work, we focus on a fractional differential equation in Riesz form discretized by a polynomial B-spline collocation method. For an arbitrary polynomial degree $p$, we show that the resulting coefficient matrices possess a Toeplitz-like structure. We investigate their spectral properties via their symbol and we prove that, like for second order differential problems, also in this case the given matrices are ill-conditioned both in the low and high frequencies for large $p$. More precisely, in the fractional scenario the symbol has a single zero at $0$ of order $\alpha$, with $\alpha$ the fractional derivative order that ranges from $1$ to $2$, and it presents an exponential decay to zero at $\pi$ for increasing $p$ that becomes faster as $\alpha$ approaches $1$. This translates in a mitigated conditioning in the low frequencies and in a deterioration in the high frequencies when compared to second order problems. Furthermore, the derivation of the symbol reveals another similarity of our problem with a classical diffusion problem. Since the entries of the coefficient matrices are defined as evaluations of fractional derivatives of the B-spline basis at the collocation points, we are able to express the central entries of the coefficient matrix as inner products of two fractional derivatives of cardinal B-splines. Finally, we perform a numerical study of the approximation behavior of polynomial B-spline collocation. This study suggests that, in line with non-fractional diffusion problems, the approximation order for smooth solutions in the fractional case is $p+2-\alpha$ for even $p$, and $p+1-\alpha$ for odd $p$.