Pad\'e-parametric FEM approximation for fractional powers of elliptic operators on manifolds

TL;DR

This paper introduces an efficient, robust numerical method for approximating fractional powers of elliptic operators on 2D manifolds using Padé approximants, with rigorous error bounds and exponential convergence.

Contribution

The paper develops a novel Padé-parametric finite element approach for fractional elliptic operators on manifolds, providing sharp error analysis and demonstrating exponential convergence.

Findings

Method is robust for all fractional powers between 0 and 1.

Algorithm achieves exponential convergence with respect to the number of solves.

Numerical tests confirm theoretical error bounds and efficiency.

Abstract

This paper focuses on numerical approximation for fractional powers of elliptic operators on $2$-d manifolds. Firstly, parametric finite element method is employed to discretize the original problem. We then approximate fractional powers of the discrete elliptic operator by the product of rational functions, each of which is a diagonal Pad\'e approximant for corresponding power function. Rigorous error analysis is carried out and sharp error bounds are presented which show that the scheme is robust for $\alpha\rightarrow 0^+$ and $\alpha \rightarrow 1^-$. The cost of the proposed algorithm is solving some elliptic problems. Since the approach is exponentially convergent with respect to the number of solves, it is very efficient. Some numerical tests are given to confirm our theoretical analysis and the robustness of the algorithm.