Numerical Approximation of Fractional Powers of Elliptic Operators

We present and study a novel numerical algorithm to approximate the action of $T^\beta:=L^{-\beta}$ where $L$ is a symmetric and positive definite unbounded operator on a Hilbert space $H_0$. The numerical method is based on a representation formula for $T^{-\beta}$ in terms of Bochner integrals involving $(I+t^2L)^{-1}$ for $t\in(0,\infty)$. To develop an approximation to $T^\beta$, we introduce a finite element approximation $L_h$ to $L$ and base our approximation to $T^\beta$ on $T_h^\beta:= L_h^{-\beta}$. The direct evaluation of $T_h^{\beta}$ is extremely expensive as it involves expansion in the basis of eigenfunctions for $L_h$. The above mentioned representation formula holds for $T_h^{-\beta}$ and we propose three quadrature approximations denoted generically by $Q_h^\beta$. The two results of this paper bound the errors in the $H_0$ inner product of $T^\beta-T_h^\beta\pi_h$ and $T_h^\beta-Q_h^\beta$ where $\pi_h$ is the $H_0$ orthogonal projection into the finite element space. We note that the evaluation of $Q_h^\beta$ involves application of $(I+(t_i)^2L_h)^{-1}$ with $t_i$ being either a quadrature point or its inverse. Efficient solution algorithms for these problems are available and the problems at different quadrature points can be straightforwardly solved in parallel. Numerical experiments illustrating the theoretical estimates are provided for both the quadrature error $T_h^\beta-Q_h^\beta$ and the finite element error $T^\beta-T_h^\beta\pi_h$.