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arXiv · 1903.10409

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

Markus Faustmann·Jens Markus Melenk·Dirk Praetorius
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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 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. Key to the analysis of the adaptive algorithm are local inverse estimates for the fractional Laplacian.

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Markus Faustmann, Jens Markus Melenk, Dirk Praetorius. 2019-03-25. Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian. https://arxiv.org/abs/1903.10409

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