The Approximation of Parabolic Equations Involving Fractional Powers of Elliptic Operators

TL;DR

This paper develops a numerical method for approximating solutions to parabolic equations involving fractional powers of elliptic operators, using contour integrals, sinc quadratures, and finite element methods, with proven error estimates.

Contribution

It introduces a novel numerical approach combining contour integral representation, sinc quadrature, and finite element approximation for fractional elliptic operators, with rigorous error analysis.

Findings

Error estimates in L^2() norm for the approximation.

Numerical experiments demonstrating the effectiveness of the method.

Efficient computation of fractional powers of elliptic operators.

Abstract

We study the numerical approximation of a time dependent equation involving fractional powers of an elliptic operator $L$ defined to be the unbounded operator associated with a Hermitian, coercive and bounded sesquilinear form on $H^1_0(\Omega)$. The time dependent solution $u(x,t)$ is represented as a Dunford Taylor integral along a contour in the complex plane. The contour integrals are approximated using sinc quadratures. In the case of homogeneous right-hand-sides and initial value $v$, the approximation results in a linear combination of functions $(z_qI-L)^{-1}v\in H^1_0(\Omega)$ for a finite number of quadrature points $z_q$ lying along the contour. In turn, these quantities are approximated using complex valued continuous piecewise linear finite elements. Our main result provides $L^2(\Omega)$ error estimates between the solution $u(\cdot,t)$ and its final approximation. Numerical results illustrating the behavior of the algorithms are provided.