REGULARITY AND NUMERICAL APPROXIMATION OF FRACTIONAL ELLIPTIC DIFFERENTIAL EQUATIONS ON COMPACT METRIC GRAPHS
Journal article, 2024

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.

Author

David Bolin

King Abdullah University of Science and Technology (KAUST)

Mihaly Kovacs

Chalmers, Mathematical Sciences

University of Gothenburg

Vivek Kumar

Bangalore Ctr

Alexandre B. Simas

King Abdullah University of Science and Technology (KAUST)

Mathematics of Computation

0025-5718 (ISSN) 1088-6842 (eISSN)

Vol. 93 349 2439-2472

Nonlocal deterministic and stochastic differential equations: analysis and numerics

Swedish Research Council (VR) (2017-04274), 2019-01-01 -- 2021-12-31.

Subject Categories

Computational Mathematics

Mathematical Analysis

DOI

10.1090/mcom/3929

More information

Latest update

7/20/2024