This study is motivated by the recent achievements in fractional calculus and its numerous applications related to anomalous (super) diffusion. Let us consider a fractional power of a self-adjoint elliptic operator introduced through its spectral decomposition. It is self-adjoint but nonlocal. The nonlocal problems are computationally expensive. Several different techniques were recently proposed to localize the nonlocal elliptic operator, thus increasing the space dimension of the original computational domain. 

An alternative approach [1,2,3] is discussed in this talk. Let $\cal A$ be a properly scaled symmetric and positive definite (SPD) sparse  matrix, arising from finite element or finite difference discretization of the initial (standard, local) diffusion problem. A method for solving algebraic systems of linear equations involving $\cal A^\alpha$, $0 < \alpha < 1$, is presented. The solution methods are based on best uniform rational approximations (BURA) of the scalar function $t^{\alpha}$, $0\le t\le 1$. The method has exponential convergence rate with respect to the degree of rational approximation $k$. The error estimates of the last variant of BURA methods are robust with respect to the spectral condition number $\kappa (\cal A)$. A stabilized modification of the Remez algorithm is developed to compute the BURA of $t^{\alpha}$. Although the fractional power of $\cal A$ is a dense matrix, the algorithm has complexity of order $O(N)$, where $N$ is the number of unknowns. At this point we assume that some solver of optimal complexity (say multigrid or multilevel) is used for the involved systems with matrices ${\cal A} + d_j \cal I$, $d_j \ge 0$, $j=1, \dots, k$. The comparative numerical tests confirm the advantages of the BURA method.

Acknowledgement: This research has been partially supported by the Bulgarian NSF Grant DN12/1.