## Journal of Applied Mathematics

### A Novel Characteristic Expanded Mixed Method for Reaction-Convection-Diffusion Problems

#### Abstract

A novel characteristic expanded mixed finite element method is proposed and analyzed for reaction-convection-diffusion problems. The diffusion term $\nabla ·\left(a\left(\mathbf{x},t\right)\nabla u\right)$ is discretized by the novel expanded mixed method, whose gradient belongs to the square integrable space instead of the classical $\mathbf{H}\left(\text{d}\text{i}\text{v};\mathrm{\Omega }\right)$ space and the hyperbolic part $d\left(\mathbf{x}\right)\left(\partial u/\partial t\right)+\mathbf{c}\left(\mathbf{x},t\right)·\nabla u$ is handled by the characteristic method. For a priori error estimates, some important lemmas based on the novel expanded mixed projection are introduced. The fully discrete error estimates based on backward Euler scheme are obtained. Moreover, the optimal a priori error estimates in ${L}^{\mathrm{2}}$- and ${H}^{\mathrm{1}}$-norms for the scalar unknown $u$ and a priori error estimates in $\left({L}^{\mathrm{2}}{\right)}^{\mathrm{2}}$-norm for its gradient $\lambda$ and its flux $\sigma$ (the coefficients times the negative gradient) are derived. Finally, a numerical example is provided to verify our theoretical results.

#### Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 683205, 11 pages.

Dates
First available in Project Euclid: 14 March 2014

https://projecteuclid.org/euclid.jam/1394807924

Digital Object Identifier
doi:10.1155/2013/683205

Mathematical Reviews number (MathSciNet)
MR3039735

Zentralblatt MATH identifier
1266.65197

#### Citation

Liu, Yang; Li, Hong; Gao, Wei; He, Siriguleng; Fang, Zhichao. A Novel Characteristic Expanded Mixed Method for Reaction-Convection-Diffusion Problems. J. Appl. Math. 2013 (2013), Article ID 683205, 11 pages. doi:10.1155/2013/683205. https://projecteuclid.org/euclid.jam/1394807924

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