Regularity and numerical approximation of fractional elliptic differential equations on compact metric graphs
Mihály Kovács (Pázmány Péter Catholic University), David Bolin, Vivek Kumar, Alexandre B. Simas
The fractional differential equation \(L^\beta u = f\) posed on a compact metric graph is considered, where \(\beta>0\) and \(L = \kappa^2 - \nabla(a\nabla)\) 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 \(L_2(\Gamma\times \Gamma)\)-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 \({L = \kappa^2 - \Delta, \kappa>0}\) are performed to illustrate the results.