Convection Diffusion Equation Characteristic Finite Element Method

Two low order nonconforming rectangular finite elements are applied to the convection-diffusion equation on anisotropic meshes in this paper. Firstly, a nonconforming rectangular finite elements is applied to the convection-diffusion equation with a characteristic finite element scheme. Under anisotropic meshes, when the solution is appropriately smooth, the O(h~2) order error estimate with respect to the space is obtained which is one order higher than the expanded characteristic-mixed finite element scheme. Secondly, we apply a non-conforming finite element to the same equation with the expanded characteristic-mixed finite element scheme on anisotropic meshes with respect to the space. The method is a combination of characteristic approximation to handle the convection part in time and an expanded mixed finite element spatial approximation to deal with the diffusion part. In the process, the anisotropic interpolation operator is used instead of the elliptic projection as in the previous literature. When the requirement of the exact solution's regularity is lower, the optimal estimates are obtained as the same as the previous literature for the conforming finite element under regular meshes.