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.
A posteriori error estimates with point sources in fractional Sobolev spaces
Universität Stuttgart（斯图加特大学）；Instituto de Matemática Aplicada del Litoral (CONICET-UNL) and Facultad de Ingeniería Química, Universidad Nacional del Litoral（国立里奥内格拉学院应用数学研究所（CONICET-UNL）和化学工程学院）；Dipartimento di Matematica, Unversità degli Studi di Milano（米兰大学数学系）
Optimal $L^\infty(L^2)$ and $L^1(L^2)$ a posteriori error estimates for the fully discrete approximations of time fractional parabolic differential equations
Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver
Brown University（布朗大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
