Numerical Approximation of Fractional Powers of Regularly Accretive Operators

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 $L^2(Ω)$, we approximate $A^{-β}$ for $β\in (0,1)$. The fractional powers are defined in terms of the so-called Balakrishnan integral formula. Given a finite element approximation space $\mathbb V_h\subset \mathbb V$, $A^{-β}$ is approximated by $A_h^{-β}π_h$ where $A_h$ is the operator associated with the form $A(\cdot,\cdot)$ restricted to $\mathbb V_h$ and $π_h$ is the $L^2(Ω)$-projection onto $\mathbb V_h$. We first provide error estimates for $(A^β-A_h^βπ_h)f$ 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 $A_h^βπ_h f$. Finally, the results of numerical computations illustrating the proposed method are given.

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