Numerical approximation of fractional powers of regularly accretive operators

Abstract

We study the numerical approximation of fractional powers of accretive operators in this paper. Namely, if A is the accretive operator asso- ciated with an accretive sesquilinear form A(·,·) defined on a Hilbert space V contained in L 2 (), we approximate A −� for � 2 (0,1). The fractional powers are defined in terms of the so-called Balakrishnan integral formula. Given a finite element approximation space VhV, A −� is approximated by A −� hh where Ah is the operator associated with the form A(·,·) restricted to Vh andh is the L 2 ()-projection onto Vh. We first provide error esti- mates for (AA � �h)f in Sobolev norms with index in (0,1) for appropriate f. These results depend on elliptic regularity properties of variational solu- tions involving the form A(·,·) and are valid for the case of less than full elliptic regularity. We also construct and analyze an exponentially convergent SINC quadrature approximation to the Balakrishnan integral defining A � �hf. Fi- nally, the results of numerical computations illustrating the proposed method are given.