Mixed Finite Element Methods For Two Classes Of Time Fractional Partial Differential Equations

In this paper, a kind of novel mixed finite element method is studied and analyzed to solve nonlinear coupled reaction-diffusion system with time fractional derivatives and convection-diffusion-wave equation with multi-term γth-order (1<γ< 2) time fractional derivatives. Compared with classical mixed element methods, the studied one has the following advantages:it is free of LBB condition; The computational complexity can be greatly reduced; The complex finite element space H(div) can be replaced by (L2(Ω))2 space.Firstly, a novel mixed element method is presented to numerically solve one-dimensional nonlinear temporal fractional coupled reaction-diffusion system. The spatial direction can be approximated by mixed element method, the integer time derivative and Caputo time fractional derivative can be discretized by second-order backward-Euler method and L1-formula, respectively, the stability and the a priori error estimates in L2-norm for fully discrete scheme. Finally, a numerical example is solved by the considered mixed element method, which test the theoretical results.Secondly, the multi-dimensional convection-diffusion-wave equation with multi-term γth-order (1<γ< 2) time fractional derivatives is solved by applying a new mixed element method. The Crank-Nicolson scheme in time is considered and novel mixed element method in space is used, the stability for fully discrete scheme is proved and the a priori error analysis for both the auxiliary variable in L2-norm and the unknown function u in H1-norm.