Regularity and numerical approximation of fractional elliptic differential equations on compact metric graphs

Abstract

The fractional differential equation L β u = f posed on a compact metric graph is considered, where β > 1 /4 and L = κ − d dx (H d dx ) is a secondorder elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients κ, H. We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when f is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power L −β . For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the L 2 (Γ×Γ)error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for the example L = κ 2 − ∆, κ > 0 are performed to illustrate the theoretical results.