Pencho Marinov

Stanislav Harizanov, Raytcho Lazarov, Pencho Marinov et al.

In this paper we consider efficient algorithms for solving the algebraic equation ${\mathcal A}^α{\bf u}={\bf f}$, $0< α<1$, where ${\mathcal A}$ is a symmetric and positive definite matrix obtained form finite difference or finite element approximations of second order elliptic problems in ${\mathbb R}^d$, $d=1,2,3$. The method is based on the best uniform rational approximation of the function $t^{β-α}$ for $0 < t \le 1$ and natural $β$, and the assumption that one has at hand an efficient method (e.g. multigrid, multilevel, or other fast algorithm) for solving equations like $({\mathcal A} +c {\mathcal I}){\bf u}= {\bf f}$, $c \ge 0$. The provided numerical experiments on model problems with ${\mathcal A}$ obtained by finite element approximation of elliptic equations in one and three spacial dimensions confirm the efficiency of the proposed algorithms.

Stanislav Harizanov, Raytcho Lazarov, Pencho Marinov et al.

We discuss, study, and compare experimentally three methods for solving the system of algebraic equations $\mathbb{A}^α\bf{u}=\bf{f}$, $0< α<1$, where $\mathbb{A}$ is a symmetric and positive definite matrix obtained from finite difference or finite element approximations of second order elliptic problems in $\mathbb{R}^d$, $d=1,2,3$. The first method, introduced by Harizanov et.al, based on the best uniform rational approximation (BURA) $r_α(t)$ of $t^{1-α}$ for $0 \le t \le 1$, is used to get the rational approximation of $t^{-α}$ in the form $t^{-1}r_α(t)$. Here we develop another method, denoted by R-BURA, that is based on the best rational approximation $r_{1-α}(t)$ of $t^α$ on the interval $[0,1]$ and approximates $t^{-α}$ via $r^{-1}_{1-α}(t)$. The third method, introduced and studied by Bonito and Pasciak, is based on an exponentially convergent quadrature scheme for the Dundord-Taylor integral representation of the fractional powers of elliptic operators. All three methods reduce the solution of the system $\mathbb{A}^α\bf{u}=\bf{f}$ to solving a number of equations of the type $(\mathbb{A} +c\mathbb{I})\bf{u}= \bf{f}$, $c \ge 0$. Comprehensive numerical experiments on model problems with $\mathbb A$ obtained by approximation of elliptic equations in one and two spatial dimensions are used to compare the efficiency of these three algorithms depending on the fractional power $α$. The presented results prove the concept of the new R-BURA method, which performs well for $α$ close to $1$ in contrast to BURA, which performs well for $α$ close to $0$. As a result, we show theoretically and experimentally, that they have mutually complementary advantages.