A mixed element method is introduced to solve Darcy-Forchheimer equation, in which the velocity and pressure are approximated by mixed element such as RaviartThomas, Brezzi-Douglas-Marini element. We establish the existence and uniqueness of the problem. Error estimates are presented based on the monotonicity owned by the Forchheimer term. An iterative scheme is given for practical computation. The numerical experiments using the lowest order Raviart-Thomas (RT 0 ) mixed element show that the convergence rates of our method are in agree with the theoretical analysis.

A new characteristic finite element scheme is presented for convection-diffusion problems. It is of second order accuracy in time increment, symmetric, and unconditionally stable. Optimal error estimates are proved in the framework of L 2 -theory. Numerical results are presented for two examples, which show the advantage of the scheme.

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We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.

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