2017/05/27 by Acosta, Gabriel, Bersetche, Francisco M., Borthagaray, Juan Pablo
#35R11 #65M60 #65R20 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.1705.09815
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.