AFEM for the fractional Laplacian
A talk in the BI.discrete Workshop series by
Markus Melenk from TU Wien
| Abstract: | Jens Markus Melenk (TU Wien), Markus Faustmann (TU Wien), Maryam Parvizi (Uni Hannover), Dirk Praetorius (TU Wien) For the discretization of the integral fractional Laplacian $(-\Delta)^s$, $0 < s < 1$, based on piecewise linear functions, we present and analyze a reliable weighted residual a posteriori error estimator. In order to compensate for a lack of $L^2$ -regularity of the residual in the regime $3/4 < s < 1$, this weighted residual error estimator includes as an additional weight a power of the distance from the mesh skeleton. We prove optimal convergence rates for an h-adaptive algorithm driven by this error estimator in the framework of [CFPP14]. Key to the analysis of the adaptive algorithm are novel local inverse estimates for the fractional Laplacian. These local inverse estimates have further applications. For example, they underlie the proof that multilevel diagonal scaling leads to uniformly bounded condition numbers even in the presence of locally refined meshes. In the second part of the talk, we will present one such optimal multilevel diagonal scaling preconditioner. [CFPP14] C. Carstensen, M. Feischl, M. Page, and D. Praetorius, Axioms of adaptivity, Comput. Math. Appl. 67 (2014), no. 6, 1195–1253. [FMP19] M. Faustmann, J.M. Melenk, M. Parvizi, On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion, M2AN, 55 (2021), 595–625 [FMP21] M. Faustmann, J.M. Melenk, D. Praetorius, Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian, Math. Comp. 90 (2021), 1557– 1587 Attendance is only possible after registration with the organizers and with 3G-certificate. Within the CRC this talk is associated to the project(s): A7, B7 |