Numerical approximation of fractional powers of elliptic operators

We present and study a novel numerical algorithm to approximate the action of T β := L − β T^β :=L-β where L L is a symmetric and positive definite unbounded operator on a Hilbert space H 0 H_0 . The numerical method is based on a representation formula for T − β T-β in terms of Bochner integrals involving ( I + t 2 L ) − 1 (I+t^2L)⁻¹ for t ∈ ( 0 , ∞ ) t∈ (0,∞ ) . To develop an approximation to T β T^β , we introduce a finite element approximation L h L_h to L L and base our approximation to T β T^β on T h β := L h