In this paper, we propose a stabilizer free spatial weak Galerkin (SFSWG) finite element method for solving time-dependent convection diffusion equations based on weak form Eq. (4). SFSWG method in spatial direction and Euler difference operator Eq. (37) in temporal direction are used. The main reason for using the SFSWG method is because of its simple formulation that makes this algorithm more efficient and its implementation easier. The optimal rates of convergence of and in a discrete and norms, respectively, are obtained under certain conditions if polynomial spaces are used in the SFSWG finite element method. Numerical experiments are performed to verify the effectiveness and accuracy of the SFSWG method.
Al-TaweelAHussainSWangX. A stabilizer free weak Galerkin finite element method for parabolic equation. Journal of Computational and Applied Mathematics. 2021; 392: 113373.
2.
KimMYShinDW. A high order discontinuous Galerkin method with skeletal multipliers for convection-diffusion-reaction problems. Computer Methods in Applied Mechanics and Engineering. 2019; 343: 207–233.
3.
StynesM. Finite volume methods for convection-diffusion problems. Journal of Computational and Applied Mathematics. 1995; 63(1-3): 83–90.
4.
XieSZhuPWangX. Error analysis of weak Galerkin finite element methods for time-dependent convection-diffusion equations. Applied Numerical Mathematics. 2019; 137: 19–33.
5.
HussainSMalluwawaduNZhuP. A weak Galerkin finite element method for the second order elliptic problem with mixed boundary condition. J Appl Anal Comput. 2018; 8(5): 1452–1463.
6.
MuLWangJWangYYeX. A computational study of the weak Galerkin method for second-order elliptic equations. Numerical Algorithms. 2013; 63(4): 753–777.
7.
MalluwawaduNHussainS. A weak Galerkin method for second order elliptic problems with polynomial reduction. Journal of Applied Analysis & Computation. 2019; 9(2): 655–670.
8.
CheichanMSKashkoolHAGaoF. A weak galerkin finite element method for solving nonlinear convection-diffusion problems in one dimension. International Journal of Applied and Computational Mathematics. 2019; 5(2): 1–15.
9.
CheichanMSKashkoolHAGaoF. A weak Galerkin finite element method for solving nonlinear convection-diffusion problems in two dimensions. Applied Mathematics and Computation. 2019; 354: 149–163.
10.
Al-TaweelAHussainSWangXJonesB. A P0–P0 weak Galerkin finite element method for solving singularly perturbed reaction-diffusion problems. Numerical Methods for Partial Differential Equations. 2020; 36(2): 213–227.
11.
MuLWangJYeX. Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes. Numerical Methods for Partial Differential Equations. 2014; 30(3): 1003–1029.
12.
MuLWangJYeX. weak Galerkin finite element method for the Helmholtz equation with large wave number on polytopal meshes. Numerical Methods for Partial Differential Equations. 2015; 35(3): 1228–1255.
13.
HusseinAJKashkoolHA. Weak Galerkin finite element method for solving one-dimensional coupled Burgers’ equations. Journal of Applied Mathematics and Computing. 2020; 63(1): 265–293.
14.
YeXZhangS. A weak Galerkin finite element method for p-laplacian problem. East Asian Journal On Applied Mathematics. 2021; 11(2): 219–233.
15.
MuLWangJYeXZhangS. A weak Galerkin finite element method for the Maxwell equations. Journal of Scientific Computing. 2015; 65(1): 363–386.
16.
YeXZhangS. A stabilizer-free weak Galerkin finite element method on polytopal meshes. Journal of Computational and Applied Mathematics. 2020; 371: 112699.
17.
Al-TaweelAWangX. A note on the optimal degree of the weak gradient of the stabilizer free weak Galerkin finite element method. Applied Numerical Mathematics. 2020; 150: 444–451.
18.
Di PietroDAErnA. Mathematical aspects of discontinuous Galerkin methods. vol. 69. Springer Science & Business Media; 2011.
19.
WangJYeX. A weak Galerkin mixed finite element method for second order elliptic problems. Mathematics of Computation. 2014; 83(289): 2101–2126.
20.
WangJYeX. A weak Galerkin finite element method for second-order elliptic problems. Journal of Computational and Applied Mathematics. 2013; 241: 103–115.
