Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian

Faustmann, Markus; Melenk, Jens; Praetorius, Dirk

doi:10.1090/mcom/3603

Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian
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by Markus Faustmann, Jens Markus Melenk and Dirk Praetorius HTML | PDF

Math. Comp. 90 (2021), 1557-1587 Request permission

Abstract:

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 the 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. Key to the analysis of the adaptive algorithm are local inverse estimates for the fractional Laplacian.

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