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 $\nabla ·(a(\mathbf{x},t)\nabla u)$ is discretized by the novel expanded mixed method, whose gradient belongs to the square integrable space instead of the classical $\mathbf{H}(\text{d}\text{i}\text{v};\mathrm{\Omega })$ space and the hyperbolic part $d(\mathbf{x})(\partial u/\partial t)+\mathbf{c}(\mathbf{x},t)·\nabla u$ 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 $(L^{\mathrm{2}}{)}^{\mathrm{2}}$-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.