Numerical Approximation of Fractional Powers of Regularly Accretive Operators

TL;DR

This paper develops and analyzes a numerical method for approximating fractional powers of accretive operators using finite element spaces, error estimates, and sinc quadrature, with validation through numerical experiments.

Contribution

It introduces a new finite element-based approximation scheme for fractional accretive operators, including error analysis and an exponentially convergent quadrature method.

Findings

Error estimates in Sobolev norms for the approximation

Construction of an exponentially convergent sinc quadrature

Numerical experiments confirming the theoretical results

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.