Positivity and discretization of Fredholm integral operators

TL;DR

This paper establishes conditions under which Fredholm integral operators and their discretizations maintain positivity, analyzing various numerical methods and their impact on eigenvalues relevant to nonlinear operators.

Contribution

It provides new sufficient conditions for positivity preservation in vector-valued Fredholm operators and their discretizations, including Nyström and projection methods.

Findings

Nyström methods preserve positivity.

Certain projection methods maintain positivity.

Quadratic splines and sinc-collocation violate positivity.

Abstract

We provide sufficient conditions for vector-valued Fredholm integral operators and their commonly used spatial discretizations to be positive in terms of an order relation induced by a corresponding order cone. It turns out that reasonable Nystr\"om methods preserve positivity. Among the projection methods, persistence is obtained for the simplest ones based on polynomial, piecewise linear or specific cubic interpolation (collocation), as well as for piecewise constant basis functions in a Bubnov-Galerkin approach. However, for semi-discretizations using quadratic splines or $\sinc$-collocation we demonstrate that positivity is violated. Our results are illustrated in terms of eigenpairs for Krein-Rutman operators and form the basis of corresponding investigations for nonlinear integral operators.