# Full-Discrete Weak Galerkin Finite Element Method for Solving Diffusion-Convection Problem.

### Abstract

This paper applied and analyzes full discrete weak Galerkin (WG) finite element method for non steady two dimensional convection-diffusion problem on conforming polygon. We approximate the time derivative by backward finite difference method and the elliptic form by WG finite element method. The main idea of WG finite element methods is the use of weak functions and their corresponding discrete weak derivatives in standard weak form of the model problem. The theoretical evidence proved that the error estimate in norm, the properties of the bilinear form, (v-elliptic and continuity), stability, and the energy conservation law.

### References

[2] C. Johnson, Numerical solution of partial differential equations by the finite element method, Cambridge University Press,(1987).

[3] Hashim A. Kashkool, Yahea H. Saleem and Ghazi A. Muften, Error estimate of discontinuous Galerkin finite element method for space time discretization of Convection â€“Diffusion problem, IJPRET,2013; Volume 2(4):25-50.

[4] Hashim A. Kashkool, Upwind type finite element method for nonlinear Convection-Diffusion problem and applications to numerical reservoir simulation, school of Mathematical Sciences, Nankai Universty, China (2002).

[5] J.T. Oden, and T.N. Reddy, An introduction to mathematical theory of finite element, John Wiley and Sons Inc., (1976)

[6] J. Wang and X.Ye, A weak Galerkin finite element method for second-order elliptic problems, arXiv:1104.2897v1,(2011).

[7] K. Baba and M. Tabata, On a conservation upwind finite element scheme for convection- diffution equation, R.A.I.R.O., Numerical analysis, 15, no. 1, pp.

3-25, (1981 ).

[8] L. Mu, J. Wang, X. Ye, Weak Galerkin finite element method on polytopal meshes, arXiv:1204.3655.

[9] L. Mu, J. Wang, X. Ye, A weak Galerkin finite element method with polynomial reduction, arXiv:1304.6481v1,(2013).

[10] Mu, L., Wang, J., Wei, G., Ye., X., Zhao, S., A weak Galerkin finite element method for the elliptic interface problem, arXiv:1201.6438v2,(2012).

[11] Mu, L., Wang, J., Wang, Y., Ye., X., A weak Galerkin mixed finite element method for Biharmonic equations, arXiv:1210.3818,(2012).

[12] Mu, L., Wang, J., Ye., X., Zhang, S., A -weak Galerkin finite element method for the biharmonic equation, arXiv:1212.0250,(2012)

[13] Mu, L., Wang, J., Ye., X., Zhao, S., Numerical studies on the weak Galerkin method for Helmholtz equation with large wave number, arXiv:1111.0671,(2011).

[14] P. G. Ciarlet, The finite element method for elliptic problems, Amesterdam. North Holland, 1978.

[15] Qiaoluan H. LI and J. Wang, Weak Galerkin finite element methods for parabolic equations, arXiv:1212.3637v2.(2013).

[16] Quarteroni and A.valli, Numerical approximation of partial differential equations, ISBN 3-540-5711-6, Springer-Verlag Berlin Heidelbery-NewYourk,(1997).

[17] S. Sun and M. F. Wheeler, Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. NumÃ©r. Anal. 43 (2005), 195â€“219.

[18] Uwe Risch, An upwind finite element method for singularly perturbed elliptic problems and local estimates in the Lâˆž-norm, MAN, 34, no. 2, pp. 235-264, (1990).

[19] V.Thomee, Galerkin finite element for parabolic problem, Springer-Verlag, Berlin,1054,(1984)

*JOURNAL OF ADVANCES IN MATHEMATICS*,

*13*(4), 7733-7345. https://doi.org/10.24297/jam.v13i4.6312

Authors retain the copyright of their manuscripts, and all Open Access articles are distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided that the original work is properly cited.