A splitting scheme to solve an equation for fractional powers of elliptic operators

TL;DR

This paper introduces a stable numerical splitting scheme for solving equations involving fractional powers of elliptic operators, using finite differences and auxiliary pseudo-parabolic equations, with demonstrated numerical effectiveness.

Contribution

The paper develops an unconditionally stable vector additive splitting scheme for fractional elliptic equations, enhancing computational stability and efficiency.

Findings

Scheme is unconditionally stable.

Numerical results confirm accuracy and efficiency.

Splitting method simplifies complex fractional problems.

Abstract

An equation containing a fractional power of an elliptic operator of second order is studied for Dirichlet boundary conditions. Finite difference approximations in space are employed. The proposed numerical algorithm is based on solving an auxiliary Cauchy problem for a pseudo-parabolic equation. Unconditionally stable vector additive schemes (splitting schemes) are constructed. Numerical results for a model problem in a rectangle calculated using the splitting with respect to spatial variables are presented.