We develop a novel a posteriori error estimator for the $L^2$ error committed by the finite element discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes advantage of the semi-discretization scheme using rational approximations which allow to reformulate the fractional problem into a family of non-fractional parametric problems. The estimator involves applying the implicit Bank-Weiser error estimation strategy to each parametric non-fractional problem and reconstructing the fractional error through the same rational approximation used to compute the solution to the original fractional problem. In addition we propose an algorithm to adapt both the finite element mesh and the rational scheme in order to balance the discretization errors. We provide several numerical examples in both two and three-dimensions demonstrating the effectivity of our estimator for varying fractional powers and its ability to drive an adaptive mesh refinement strategy.

翻译：我们针对分数阶拉普拉斯算子有限元离散解的$L^2$误差，提出了一种新型后验误差估计子。该估计子利用基于有理近似的半离散化方案，将分数阶问题转化为一族非分数阶参数化问题。估计过程涉及将隐式Bank-Weiser误差估计策略应用于每个参数化非分数阶问题，并通过与求解原分数阶问题相同的有理近似重构分数阶误差。此外，我们提出了一种算法来自适应调整有限元网格和有理近似方案，以平衡离散化误差。我们提供了二维和三维空间中的多个数值算例，验证了所提估计子在不同分数阶次下的有效性，及其驱动自适应网格细化策略的能力。