Regularity and numerical approximation of fractional elliptic differential equations on compact metric graphs
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- by David Bolin, Mihály Kovács, Vivek Kumar and Alexandre B. Simas;
- Math. Comp. 93 (2024), 2439-2472
- DOI: https://doi.org/10.1090/mcom/3929
- Published electronically: December 27, 2023
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Abstract:
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.References
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Bibliographic Information
- David Bolin
- Affiliation: Statistics Program, Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia
- MR Author ID: 918715
- ORCID: 0000-0003-2361-5465
- Email: david.bolin@kaust.edu.sa
- Mihály Kovács
- Affiliation: Department of Mathematical Sciences,Chalmers University of Technology and University of Gothenburg, SE-41296 Gothenburg, Sweden; Department of Differential Equations, Budapest University of Technology and Economics, Muegyetem rkp. 3., H-1111 Budapest, Hungary; and Faculty of Information Technology and Bionics, Pázmány Péter Catholic University, Práter utca 50/a., H-1083 Budapest, Hungary
- ORCID: 0000-0001-7977-9114
- Email: mkovacs@math.bme.hu
- Vivek Kumar
- Affiliation: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore Centre, 8th Mile Mysore Road, Bangalore, 560059, Karnataka, India
- MR Author ID: 1319257
- Email: vivekkumar\!_ra@isibang.ac.in
- Alexandre B. Simas
- Affiliation: Statistics Program, Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia
- MR Author ID: 864018
- ORCID: 0000-0003-2562-2829
- Email: alexandre.simas@kaust.edu.sa
- Received by editor(s): February 8, 2023
- Received by editor(s) in revised form: September 19, 2023
- Published electronically: December 27, 2023
- Additional Notes: The second author was supported by the Marsden Fund of the Royal Society of New Zealand (grant no. 18-UOO-143), the Swedish Research Council (VR) (grant no. 2017-04274) and the National Research, Development, and Innovation Fund of Hungary (grant no. TKP2021-NVA-02 and K-131545). The third author was supported in part by a NBHM post-doctoral fellowship from the Department of Atomic Energy (DAE), Government of India (file no. 0204/6/2022/R&D-II/5635).
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 93 (2024), 2439-2472
- MSC (2020): Primary 35R02, 35A01, 35A02, 60H15, 60H40
- DOI: https://doi.org/10.1090/mcom/3929
- MathSciNet review: 4759380