Regularity and numerical approximation of fractional elliptic differential equations on compact metric graphs
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The fractional differential equation L^β u = f posed on a compact metric graph is considered, where β>1/4 and L = κ - d/d x(Hd/d x) is a second-order 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 - Δ, κ>0 are performed to illustrate the theoretical results.
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