scholarly journals A Crank–Nicolson Finite Volume Element Method for Time Fractional Sobolev Equations on Triangular Grids

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1591
Author(s):  
Jie Zhao ◽  
Zhichao Fang ◽  
Hong Li ◽  
Yang Liu
Keyword(s):  
Finite Volume ◽  
A Priori ◽  
Stability Result ◽  
Volume Element ◽  
A Priori Error Estimate ◽  
Fully Discrete ◽  
Sobolev Equations ◽  
Finite Volume Element ◽  
Nicolson Scheme ◽  
Crank Nicolson

In this paper, a finite volume element (FVE) method is proposed for the time fractional Sobolev equations with the Caputo time fractional derivative. Based on the L1-formula and the Crank–Nicolson scheme, a fully discrete Crank–Nicolson FVE scheme is established by using an interpolation operator Ih*. The unconditional stability result and the optimal a priori error estimate in the L2(Ω)-norm for the Crank–Nicolson FVE scheme are obtained by using the direct recursive method. Finally, some numerical results are given to verify the time and space convergence accuracy, and to examine the feasibility and effectiveness for the proposed scheme.

A Priori Error Estimates of Crank-Nicolson Finite Volume Element Method for Parabolic Optimal Control Problems

2013 ◽  
Vol 5 (05) ◽  
pp. 688-704 ◽  
Author(s):  
Xianbing Luo ◽  
Yanping Chen ◽  
Yunqing Huang
Keyword(s):  
Optimal Control ◽  
Finite Volume ◽  
Optimal Control Problems ◽  
A Priori ◽  
Volume Element ◽  
Control Problems ◽  
Finite Volume Element Method ◽  
Finite Volume Element ◽  
Element Method ◽  
Crank Nicolson

AbstractIn this paper, the Crank-Nicolson linear finite volume element method is applied to solve the distributed optimal control problems governed by a parabolic equation. The optimal convergent orderO(h2+k2) is obtained for the numerical solution in a discreteL2-norm. A numerical experiment is presented to test the theoretical result.

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

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Jie Zhao ◽  
Hong Li ◽  
Zhichao Fang ◽  
Xue Bai
Keyword(s):  
Numerical Solution ◽  
Finite Volume ◽  
Burgers Equation ◽  
A Priori ◽  
Transfer Operator ◽  
Volume Element ◽  
Fully Discrete ◽  
Finite Volume Element ◽  
The Stability ◽  
Stability Results

In this article, mixed finite volume element (MFVE) methods are proposed for solving the numerical solution of Burgers’ equation. By introducing a transfer operator, semidiscrete and fully discrete MFVE schemes are constructed. The existence, uniqueness, and stability analyses for semidiscrete and fully discrete MFVE schemes are given in detail. The optimal a priori error estimates for the unknown and auxiliary variables in the L2Ω norm are derived by using the stability results. Finally, numerical results are given to verify the feasibility and effectiveness.

scholarly journals A Fully Discrete Symmetric Finite Volume Element Approximation of Nonlocal Reactive Flows in Porous Media

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Zhe Yin ◽  
Qiang Xu
Keyword(s):  
Porous Media ◽  
Finite Volume ◽  
Volume Element ◽  
Optimal Order ◽  
Element Approximation ◽  
Two Dimensional ◽  
Reactive Flows ◽  
Flows In Porous Media ◽  
Fully Discrete ◽  
Finite Volume Element

We study symmetric finite volume element approximations for two-dimensional parabolic integrodifferential equations, arising in modeling of nonlocal reactive flows in porous media. It is proved that symmetric finite volume element approximations are convergent with optimal order inL2-norm. Numerical example is presented to illustrate the accuracy of our method.

A New Finite Volume Element Formulation for the Non-Stationary Navier-Stokes Equations

2014 ◽  
Vol 6 (5) ◽  
pp. 615-636 ◽  
Author(s):  
Zhendong Luo
Keyword(s):  
Finite Element ◽  
Finite Volume ◽  
Stokes Equations ◽  
Volume Element ◽  
Navier Stokes ◽  
Element Formulation ◽  
Navier Stokes Equations ◽  
Fully Discrete ◽  
Finite Volume Element ◽  
Stationary Navier Stokes Equations

AbstractA semi-discrete scheme about time for the non-stationary Navier-Stokes equations is presented firstly, then a new fully discrete finite volume element (FVE) formulation based on macroelement is directly established from the semi-discrete scheme about time. And the error estimates for the fully discrete FVE solutions are derived by means of the technique of the standard finite element method. It is shown by numerical experiments that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the FVE method is feasible and efficient for finding the numerical solutions of the non-stationary Navier-Stokes equations and it is one of the most effective numerical methods among the FVE formulation, the finite element formulation, and the finite difference scheme.

ON THE APPROXIMATION OF INCOMPRESSIBLE MISCIBLE DISPLACEMENT PROBLEMS IN POROUS MEDIA BY MIXED AND STANDARD FINITE VOLUME ELEMENT METHODS

2013 ◽  
Vol 04 (03) ◽  
pp. 1350013 ◽  
Author(s):  
SARVESH KUMAR
Keyword(s):  
Porous Media ◽  
Finite Volume ◽  
A Priori ◽  
Nonlinear Partial Differential Equations ◽  
Coupled System ◽  
Miscible Displacement ◽  
Volume Element ◽  
Finite Volume Element ◽  
Velocity Equation ◽  
Concentration Equation

The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure–velocity equation and the concentration equation. In this paper, we present a mixed finite volume element method (FVEM) for the approximation of the pressure–velocity equation and a standard FVEM for the concentration equation. A priori error estimates in L∞(L2) are derived for velocity, pressure and concentration. Numerical results are presented to substantiate the validity of the theoretical results.

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