Finite element approximations for fractional evolution problems
It provides a numerical method for solving fractional evolution equations that model memory and long-range dispersion, but the results are incremental and lack concrete performance numbers.
The paper develops and analyzes a finite element scheme for evolution problems with fractional time and space derivatives up to order two, demonstrating its performance through numerical experiments in 1D and 2D domains.
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.