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

Abstract

The fractional differential equation L β u = f L^\beta u = f posed on a compact metric graph is considered, where β > 0 \beta >0 and L = κ 2 − ∇ ( a ∇ ) L = \kappa ^2 - \nabla (a\nabla ) is a second-order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients κ , a \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 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 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 ( Γ × Γ ) L_2(\Gamma \times \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 = κ 2 − Δ , κ > 0 {L = \kappa ^2 - \Delta , \kappa >0} are performed to illustrate the results.