REGULARITY AND NUMERICAL APPROXIMATION OF FRACTIONAL ELLIPTIC DIFFERENTIAL EQUATIONS ON COMPACT METRIC GRAPHS

Journal article, 2023

The fractional differential equation L(beta)u = f posed on a compact metric graph is considered, where beta > 0 and L = kappa(2) - del(a del ) is a second order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients kappa, a. 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-beta. 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( Gamma x Gamma )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 L = kappa(2) - del, kappa > 0 are performed to illustrate the results.