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.

中文翻译：

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积分分数拉普拉斯算子自适应方法的准最优收敛速度

对于基于分段线性函数的积分分数拉普拉斯算子 $(-\Delta)^s$, $0 < s < 1$ 的离散化，我们提出并分析了一个可靠的加权残差后验误差估计量。为了补偿 $3/4 < s < 1$ 范围内残差的 $L^2$-正则性的缺失，该加权残差估计器包括与网格骨架的距离的幂作为附加权重。我们证明了由该误差估计器驱动的 $h$ 自适应算法的最佳收敛率。自适应算法分析的关键是分数拉普拉斯算子的局部逆估计。