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 ∇·(a(x,t)∇u) is discretized by the novel expanded mixed method, whose gradient belongs to the square integrable space instead of the classical H(div;Ω) space and the hyperbolic part d(x)(∂u/∂t)+c(x,t)·∇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 L2- and H1-norms for the scalar unknown u and a priori error estimates in (L2)2-norm for its gradient λ and its flux σ (the coefficients times the negative gradient) are derived. Finally, a numerical example is provided to verify our theoretical results.