Numerical Approximation of Fractional Powers of Elliptic Operators

TL;DR

This paper develops and analyzes two numerical algorithms based on time-stepping schemes with Padé approximation for efficiently computing fractional powers of elliptic operators, with error estimates and numerical validation.

Contribution

It introduces two novel time-stepping algorithms for fractional elliptic operators using Padé approximation, including error analysis and practical implementation details.

Findings

Error estimates depend on the regularity of the data

Geometrically graded meshes improve accuracy near singularities

Numerical experiments confirm theoretical convergence rates

Abstract

In this paper, we develop and study algorithms for approximately solving the linear algebraic systems: $\mathcal{A}_h^\alpha u_h = f_h$, $ 0< \alpha <1$, for $u_h, f_h \in V_h$ with $V_h$ a finite element approximation space. Such problems arise in finite element or finite difference approximations of the problem $ \mathcal{A}^\alpha u=f$ with $\mathcal{A}$, for example, coming from a second order elliptic operator with homogeneous boundary conditions. The algorithms are motivated by the recent method of Vabishchevich, 2015, that relates the algebraic problem to a solution of a time-dependent initial value problem on the interval $[0,1]$. Here we develop and study two time stepping schemes based on diagonal Pad\'e approximation to $(1+x)^{-\alpha}$. The first one uses geometrically graded meshes in order to compensate for the singular behavior of the solution for $t$ close to $0$. The second algorithm uses uniform time stepping but requires smoothness of the data $f_h$ in discrete norms. For both methods, we estimate the error in terms of the number of time steps, with the regularity of $f_h$ playing a major role for the second method. Finally, we present numerical experiments for $\mathcal{A}_h$ coming from the finite element approximations of second order elliptic boundary value problems in one and two spatial dimensions.