A Mixed Finite Volume Element Method for Time-Fractional Reaction-Diffusion Equations on Triangular Grids

Abstract

In this article, the time-fractional reaction-diffusion equations are solved by using a mixed finite volume element (MFVE) method and the L 1 -formula of approximating the Caputo fractional derivative. The existence, uniqueness and unconditional stability analysis for the fully discrete MFVE scheme are given. A priori error estimates for the scalar unknown variable (in L 2 ( Ω ) -norm) and the vector-valued auxiliary variable (in ( L 2 ( Ω ) ) 2 -norm and H ( div , Ω ) -norm) are derived. Finally, two numerical examples in one-dimensional and two-dimensional spatial regions are given to examine the feasibility and effectiveness.

Finite volume element method with the WSGD formula for nonlinear fractional mobile/immobile transport equations

“…Remark 2.1 For the fully discrete FVE scheme (17) or equivalent formulations (18)- (19), in the practical calculations, we can obtain U 1 by using U 0 and solving (18), where U 0 = P h u 0 defined in Sect. 4.…”

“…We substitute (34) into the formulations (18) and (19) equivalent to the FVE scheme (17), take v h = Φ j (j = 1, 2, . .…”

“…However, due to the existence and complexity of fractional derivative, it is difficult to obtain the analytical solutions for most of the FDEs. Thus, many numerical methods [8][9][10][11][12][13][14][15][16][17][18][19] have been proposed and studied to solve different types of the FDEs.…”

Numerical Solution of Burgers’ Equation Based on Mixed Finite Volume Element Methods

“…e MFVE methods, also called mixed covolume methods, were first proposed by Russell [31] to solve the elliptic equation. Now, the methods have been applied to solve second-order elliptic equations [32][33][34], integrodifferential equations [35], parabolic equations [36,37], time-fractional partial differential equations [38], and so on. In this article, we introduce a flux function as an auxiliary variable, rewrite (1) as the first-order system, and construct the semidiscrete and nonlinear backward Euler fully discrete MFVE schemes by using a transfer operator.…”

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.…”