Finite element approximations for fractional evolution problems

BERSETCHE, FRANCISCO M.; ACOSTA, GABRIEL; BORTHAGARAY, JUAN PABLO

Fractional Calculus and Applied Analysis

De Gruyter

Año: 2019 vol. 22 p. 767 - 794

1311-0454

This work introduces and analyzes a finite element scheme for evolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time we consider is employed to represent memory effects, while a nonlocal differentiation operator in space accounts for long-range dispersion processes. We discuss well-posedness and obtain regularity estimates for the evolution problems under consideration. The discrete scheme we develop is based on piecewise linear elements for the space variable and a convolution quadrature for the time component. We illustrate the method?s performance with numerical experiments in one- and two-dimensional domains.