Numerical solving unsteady space-fractional problems with the square root of an elliptic operator

TL;DR

This paper develops and analyzes numerical methods for solving unsteady space-fractional problems involving the square root of an elliptic operator, using finite element approximation and regularized time schemes.

Contribution

It introduces a novel combination of finite element spatial discretization with regularized two-level and three-level schemes for time integration of space-fractional equations.

Findings

Second-order accuracy in time achieved with regularized schemes

Numerical experiments validate the effectiveness of the methods

Applicable to problems with convective terms

Abstract

An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the square root of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The scheme of the second-order accuracy in time is based on a regularization of the three-level explicit Adams scheme. More general problems for the equation with convective terms are considered, too. The results of numerical experiments are presented for a model two-dimensional problem.