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A Galerkin finite element method (FEM) is developed for nonlinear Poisson–Nernst–Planck (PNP) equations. Based on the combination technique of the element's interpolation and Ritz projection together with the special mean value approach, the superclose and superconvergence estimates with order O (h 2) are derived under weaker regularity requirements of the exact solution. Some numerical results are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

A new low order two-grid mixed finite element method (FEM) is developed for the nonlinear Benjamin-Bona-Mahoney (BBM) equation, in which the famous nonconforming rectangular C N Q 1 r o t element and Q 0 × Q 0 constant element are used to approximate the exact solution u and the variable p → = ∇ u t , respectively. Then, based on the special properties of these two elements and interpolation post-processing technique, the superconvergence results for u in broken H 1-norm and p → in L 2-norm are obtained for the semi-discrete and Crank-Nicolson fully-discrete schemes without the restriction between the time step τ and coarse mesh size H or the fine mesh size h , which improve the results of the existing literature. Finally, some numerical results are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]