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Abstract In this work we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker–Planck equation and the Keller–Segel equations. The two proposed schemes are first-order accurate in time, explicitly solvable, and second-order and fourth-order accurate in space, which are obtained via finite difference implementation of the classical continuous finite element method. The fully discrete schemes are proved to be positivity preserving and energy dissipative: the second-order scheme can achieve so unconditionally while the fourth-order scheme only requires a mild time step and mesh size constraint. In particular, the fourth-order scheme is the first high order spatial discretization that can achieve both positivity and energy decay properties, which is suitable for long time simulation and to obtain accurate steady state solutions. 

The classical continuous finite element method with Lagrangian Q^k basis reduces to a finite difference scheme when all the integrals are replaced by the (𝑘+1)×(𝑘+1) Gauss–Lobatto quadrature. We prove that this finite difference scheme is (𝑘+2)-th order accurate in the discrete 2-norm for an elliptic equation with Dirichlet boundary conditions, which is a superconvergence result of function values. We also give a convenient implementation for the case 𝑘=2, which is a simple fourth order accurate elliptic solver on a rectangular domain. 

We show that the fourth order accurate finite difference implementation of continuous finite element method with tensor product of quadratic polynomial basis is monotone thus satisfies the discrete maximum principle for solving a scalar variable coefficient equation −∇⋅(𝑎∇𝑢)+𝑐𝑢=𝑓 under a suitable mesh constraint.