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A nonconforming E Q 1 r o t finite element method (FEM) is studied for the nonlinear coupled predator-prey equations. The superconvergence estimates are derived for the semi-discrete and Crank-Nicolson (C-N) fully discrete schemes based on the two special properties of this element: one is that the interpolation operator is equivalent to its projection operator, and the other is that the consistency error can be estimated as order O (h 2) in the broken H 1 -norm when the exact solution belongs to H 3 (Ω) , which is one order higher than that of its interpolation error estimate. Moreover, the unique solvability of the nonlinear coupled semi-discrete scheme is certified through the Brouwer fixed point theorem analytically. On the other hand, the stability of the decoupled C-N fully discrete scheme is proved by mathematical induction, which leads to the unconditional superconvergence of order O (h 2 + τ 2) without the ratio between the time-step τ and the mesh size h. Finally, numerical examples are given to demonstrate the validity of our method. • Based on two properties of E Q 1 r o t element, we get superclose and superconvergence for the semi-discrete and C-N schemes. • The unique solvability of the nonlinear coupled semi-discrete scheme is proved by Brouwer fixed point theorem analytically. • The stability of the decoupled C-N scheme is proved by mathematical induction. • From the stability, we can eliminate the inverse inequality to get the superconvergence without time-step restriction. • The analysis and the results presented are also valid for other popular finite elements when they satisfy some conditions. [ABSTRACT FROM AUTHOR]

This paper aims to study the Galerkin finite element method (FEM) of both the semi-discrete and the fully-discrete schemes with linear triangular element for the time-dependent Joule heating problem. Through some new estimating approaches such as the different mathematical induction assumption, the mean-value technique and the special characters of this element, the superclose and superconvergence estimates for the related variables in the H 1 -norm are derived under lower regularities of the solutions of the considered problem rigorously, which simply the proof process and improve the corresponding results in the existing literature. Finally, numerical results are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, the discontinuous Galerkin method (DGM) of nonconforming Wilson element is studied for the semi-linear parabolic problem. The global superconvergence with respect to the mesh size are derived in the modified H 1 -norm for the semi-discrete scheme and two fully discrete schemes, in which the usual extrapolation and interpolation post-processing approaches are not involved, and the error estimates are one order higher than that of the traditional Galerkin finite element method (FEM). Therefore, the corresponding results in the existing literature are improved. Finally, some numerical results are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract An H 1 -Galerkin mixed finite element method (MFEM) is developed for the Benjamin–Bona–Mahony (BBM) equation with the bilinear element and zero order Raviart–Thomas element pair (Q 11 ∕ Q 10 × Q 01). Then based on the special interpolation properties of the above two elements and the mean-value technique, the supercloseness and superconvergence results of order O (h 2) for the semi-discrete scheme and of order O (h 2 + (Δ t) 2) for the Crank–Nicolson fully-discrete scheme are deduced for the original variable u in H 1 -norm and the flux p → in H (d i v , Ω) -norm, respectively. Finally, a numerical example is carried out to demonstrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

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]