NUMERICAL APPROXIMATION OF FRACTIONAL POWERS OF ELLIPTIC OPERATORS

We present and study a novel numerical algorithm to approximate the action of Tβ := L-β where L is a symmetric and positive definite unbounded operator on a Hilbert space H0. The numerical method is based on a representation formula for T-β in terms of Bochner integrals involving (I + t2L)-1 for t ∈ (0,∞). To develop an approximation to Tβ, we introduce a finite element approximation Lh to L and base our approximation to Tβ on Tβh := L-βh. The direct evaluation of Tβh is extremely expensive as it involves expansion in the basis of eigenfunctions for Lh. The above mentioned representation formula holds for T-βh and we propose three quadrature approximations denoted generically by Qβh. The two results of this paper bound the errors in the H0 inner product of Tβ -Tβh πh and Tβh -Qβh where πh is the H0 orthogonal projection into the finite element space. We note that the evaluation of Qβh involves application of (I +(ti)2Lh)-1 with ti 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 Tβh - Qβh and the finite element error Tβ - Tβh πh.