scholarly journals A two-grid mixed finite volume element method for nonlinear time fractional reaction-diffusion equations

2021 ◽  
Vol 7 (2) ◽  
pp. 1941-1970
Author(s):  
Zhichao Fang ◽  
◽  
Ruixia Du ◽  
Hong Li ◽  
Yang Liu
Keyword(s):  
Finite Volume ◽  
Matrix Theory ◽  
Reaction Diffusion ◽  
Volume Element ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Optimal Error Estimates ◽  
Fine Grid ◽  
Finite Volume Element ◽  
Fractional Reaction

<abstract><p>In this paper, a two-grid mixed finite volume element (MFVE) algorithm is presented for the nonlinear time fractional reaction-diffusion equations, where the Caputo fractional derivative is approximated by the classical $ L1 $-formula. The coarse and fine grids (containing the primal and dual grids) are constructed for the space domain, then a nonlinear MFVE scheme on the coarse grid and a linearized MFVE scheme on the fine grid are given. By using the Browder fixed point theorem and the matrix theory, the existence and uniqueness for the nonlinear and linearized MFVE schemes are obtained, respectively. Furthermore, the stability results and optimal error estimates are derived in detailed. Finally, some numerical results are given to verify the feasibility and effectiveness of the proposed algorithm.</p></abstract>

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

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 600 ◽  
Author(s):  
Jie Zhao ◽  
Hong Li ◽  
Zhichao Fang ◽  
Yang Liu
Keyword(s):  
Finite Volume ◽  
A Priori ◽  
Reaction Diffusion ◽  
Auxiliary Variable ◽  
Volume Element ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Unknown Variable ◽  
Finite Volume Element ◽  
Fractional Reaction

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.

A Stabilized Finite Volume Element Method for Stationary Stokes–Darcy Equations Using the Lowest Order

2019 ◽  
Vol 17 (08) ◽  
pp. 1950053
Author(s):  
Yanyun Wu ◽  
Liquan Mei ◽  
Meilan Qiu ◽  
Yuchuan Chu
Keyword(s):  
Finite Volume ◽  
Mass Conservation ◽  
Volume Element ◽  
Optimal Error Estimates ◽  
Irregular Domain ◽  
Finite Volume Element Method ◽  
Stabilized Finite Element Method ◽  
Finite Volume Element ◽  
Text Element ◽  
Element Method

We present a stabilized finite volume element method for the coupled Stokes–Darcy problem with the lowest order [Formula: see text] element for the Stokes region and [Formula: see text] element for the Darcy region. Based on adding a jump term of discrete pressure to the approximation equation, a discrete inf-sup condition is established for the proposed method. The optimal error estimates in the [Formula: see text]-norm for the velocity and piezometric head and in the [Formula: see text]-norm for the pressure are proved. And they are also verified through some numerical experiments. Two figures are given to show the full comparison for the local mass conservation between the proposed method and the stabilized finite element method. And this method can also be computed directly in the irregular domain according to the last experiment.

Sign in / Sign up

Export Citation Format

Share Document