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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]

A nonconforming finite element method (FEM) is developed and investigated for the coupled Poisson–Nernst–Planck (PNP) equations with low order EQ1rot element. Then, by use of the special properties of this element, that is, the interpolation operator is equivalent to its projection operator, and the consistency error estimate can reach order of O(h2) which is one order higher than that of its interpolation error estimates when the exact solution belongs to H3(Ω), the superclose estimates of order O(h2) and O(h2 + τ) in the broken H1‐norm are derived with new techniques for the semidiscrete scheme and backward Euler fully discrete scheme, respectively. Further, through employing interpolation postprocessing approach, the corresponding global superconvergence results are obtained. Finally, some numerical results are provided to confirm the theoretical analysis. It seems that our results have never been found in the existing literature. Here h and τ denote the mesh size and time step, respectively. [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]