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 associated with an accretive sesquilinear form $A(\cdot ,\cdot )$ defined on a Hilbert space $\mathbb{V}$ contained in $L^2(\Omega )$, we approximate $A^{-\beta }$ for $\beta \in (0,1)$. The fractional powers are defined in terms of the so-called Balakrishnan integral formula. Given a finite element approximation space ${\mathbb{V}}_h\subset \mathbb{V}$, $A^{-\beta }$ is approximated by $A_h^{-\beta }{\pi }_h$ where $A_h$ is the operator associated with the form $A(\cdot ,\cdot )$ restricted to ${\mathbb{V}}_h$ and ${\pi }_h$ is the $L^2(\Omega )$-projection onto ${\mathbb{V}}_h$. We first provide error estimates for $(A^{\beta }-A_h^{\beta }{\pi }_h)f$ in Sobolev norms with index in [0,1] for appropriate $f$. These results depend on elliptic regularity properties of variational solutions involving the form $A(\cdot ,\cdot )$ 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 $A_h^{\beta }{\pi }_{h}f$. Finally, the results of numerical computations illustrating the proposed method are given.

arXiv comment25 pages