High-Order Two and Three Level Schemes for Solving Fractional Powers of Elliptic Operators

In this paper we develop and investigate numerical algorithms for solving the fractional powers of discrete elliptic operators \({\mathcal A}_h^\alpha U = F\), 0 < α < 1, for F ∈ Vh with Vh a finite element or finite difference approximation space. Our goal is to construct efficient time stepping schemes for the implementation of the method based on the solution of a pseudo-parabolic problem. The second and fourth order approximations are constructed by using two- and three-level schemes. In order to increase the accuracy of approximations the geometric graded time grid is constructed which compensates the singular behavior of the solution for t close to 0. This apriori adaptive grid is compared with aposteriori adaptive grids. Results of numerical experiments are presented, they agree well with the theoretical results.