Métodos numéricos para problemas no locales de evolución

Abstract

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

The aim of this work is to study numerical approximations for evolution problems of the form C ∂ α t u + (−∆)su = f in Ω × (0, T), where (−∆)s stands for the fractional Laplacian operator in its integral form and C ∂ α t u(x, t) represents the Caputo derivative. To be more precise, (−∆)su(x) = C(n, s) p.v. ˆ Rn u(x) − u(y) |x − y| n+2s dy, and C ∂ α t u(x, t) = ( 1 Γ(k−α) ´ t 0 1 (t−r)α−k+1 ∂ ku ∂rk (x, r) dr if k − 1 < α < k, k ∈ N, ∂ ku ∂tk u(x, t) if α = k ∈ N. We deal with existence, uniqueness and regularity of solutions in the linear context (i.e. f = f(x, t)). The cases under study include fractional counterparts of the standard diffusion and wave models. Linear finite elements are used for the spatial variable and convolution quadrature techniques for handling the time fractional operator. Error bounds, uniform in the discretization parameters for values of t away from zero, are given. These results are extended to the semi-linear case with f(u) = u − u 3 appearing in the classical Allen-Cahn equations modeling phase separation for binary alloys. Additionally, the asymptotic behaviour of the solutions for s → 0 is studied in this particular context. Implementation details, particularly for the finite element method involving full fractional stiffness matrices and numerical quadratures for singular kernels, are carefully documented