An expanded mixed finite element simulation for two-sided time-dependent fractional diffusion problem

Abstract

In this paper, we consider a time-dependent diffusion problem with two-sided Riemann-Liouville fractional derivatives. By introducing a fractional-order flux as auxiliary variable, we establish the saddle-point variational formulation, based on which we employ a locally conservative mixed finite element method to approximate the unknown function, its derivative and the fractional flux in space and use the backward Euler scheme to discrete the time derivative, and thus propose a fully discrete expanded mixed finite element procedure. We prove the well-posedness and the optimal order error estimates of the proposed procedure for a sufficiently smooth solution. Numerical experiments are presented to confirm our theoretical findings.

Numerical Solutions to a Fractional Diffusion Equation Used in Modelling Dye-Sensitized Solar Cells

“…Esen et al [13] provided a finite-element basis for the time-fractional diffusion equation under Dirichlet boundary conditions. Yuan and Chen [39] solved a mixed diffusion problem featuring the two-sided Riemann–Liouville fractional derivatives by an FEM scheme. Li and Wang [21] solved time-fractional reaction–diffusion and diffusion-wave equations, using Galerkin FEM schemes [37] which enjoy unconditional stability.…”

A fully implicit finite difference scheme based on extended cubic B-splines for time fractional advection–diffusion equation

“…Proof Apply the mathematical induction to verify inequality (26). Put n = 0 in (25) which now takes the form…”

“…Zhu et al [25] derived an efficient differential quadrature scheme based on modified trigonometric cubic B-spline for the solution of 1D and 2D time fractional advection-diffusion equations. Yuan and Chen [26] presented an expanded mixed finite element method for the two-sided time-dependent fractional diffusion problem with two-sided Riemann-Liouville fractional derivatives.…”

Superconvergence of a finite element method for the time-fractional diffusion equation with a time-space dependent diffusivity

“…Numerical methods for time-fractional IBVPs with constant or time-independent diffusion parameter have received a huge amount of attention over the last decade. For such problems, several numerical methods have been proposed and analyzed, such as finite difference method [7, 19-21, 27, 36, 38], finite element method [6,32,39,41,43,44,48], discontinuous Galerkin (DG) methods [3, 4, 9-11, 31, 34], spectral method [23], and finite volume method [15,46], etc. The time-fractional IBVPs (1a)-(1b) with time-space dependent diffusivity is indeed very interesting and also practically important, and the numerical solutions of this problems were considered by a few authors only.…”