Local energy estimates for the fractional Laplacian.
Resumen:
The integral fractional Laplacian of order s∈(0,1) is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result the global convergence rate of the numerical solutions. For finite element discretizations, we derive local error estimates in the Hs-seminorm and show optimal convergence rates in the interior of the domain by only assuming meshes to be shape-regular. These estimates quantify the fact that the reduced approximation error is concentrated near the boundary of the domain. We illustrate our theoretical results with several numerical examples.
| 2022 | |
| Juan Pablo Borthagaray ha sido financiado en parte por la beca DMS-1411808 de la NSF y la beca de viaje de AMS-Simons. | |
|
Finite elements Error estimates Interior error estimates Fractional Laplacian |
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| Inglés | |
| Universidad de la República | |
| COLIBRI | |
| https://hdl.handle.net/20.500.12008/47829 | |
| Acceso abierto | |
| Licencia Creative Commons Atribución - No Comercial - Sin Derivadas (CC - By-NC-ND 4.0) |
