Second-order approximation scheme combined with H-1-Galerkin MFE method for nonlinear time fractional convection-diffusion equation

摘要

In this article, a second-order approximation scheme combined with an H-1-Galerkin mixed finite element (MFE) method for solving nonlinear convection-diffusion equation with time fractional derivative is proposed and analyzed. By introducing an auxiliary variable, a coupled system is formulated, then the spatial direction is approximated by H-1-Galerkin MFE method and the temporal fractional derivative and integer derivative are discretized by second -order weighted and shifted Grunwald difference (WSGD) formula and linearized second-order difference scheme, respectively. The optimal priori error estimates in L-2 and H-1-norm for the unknown function and the auxiliary variable with second-order convergent rate in time are obtained. Compared to the commonly used Ll-approximation with (2-alpha)th-order convergence rate, our method can arrive at the order 2 in time. What is more, compared with the standard finite element method, our method can well approximate the auxiliary variable. Finally, the detailed computational process of the studied numerical algorithm is shown and a nonlinear numerical example with calculated data and some figures is provided to verify our theoretical analysis. (C) 2016 Elsevier Ltd. All rights reserved.