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

The fractional differential equation L^βu = f posed on a compact metric graph is considered, where β > 0 and L = κ² − ∇(a∇) is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients κ, 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^−β. For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the L2(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 L = κ² − Δ, κ > 0 are performed to illustrate the results.