Numerical approximation of fractional powers of regularly accretive operators

Abstract

We study the numerical approximation of fractional powers of accretive operators in this article. Namely, if A is the accretive operator associated with a regular sesquilinear form A(.,.) defined on a Hilbert space V contained in L2(ω), we approximate A-β for β ϵ (0, 1). The fractional powers are defined in terms of the so-called Balakrishnan integral formula. Given a finite element approximation space Vh V, A-β is approximated by Ah-βh, where Ah is the operator associated with the form A(., .) restricted to Vh, and h is the L2(ω)-projection onto Vh. We first provide error estimates for (A-β-Ah-βh)f in Sobolev norms with index in [0,1] for appropriate f . These results depend on the elliptic regularity properties of variational solutions involving the form A(., .) and are valid for the case of less than full elliptic regularity. We also construct and analyse an exponentially convergent sinc quadrature approximation to the Balakrishnan integral defining Ah-βhf. Finally, the results of numerical computations illustrating the proposed method are given.