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The aim of this paper is to develop and investigate a fully-discrete scheme with conforming P 1 element for the nonlinear time-fractional Joule heating problem in which the Caputo derivative is approximated by the classical L 1 method. First, a novel superclose estimate in the H 1 -norm is derived rigorously with some new analysis techniques under low regularity of the solutions u n , ϕ n ∈ L ∞ (0 , T ; H 3 (Ω)) rather than u n ∈ L ∞ (0 , T ; H 4 (Ω)) and ϕ n ∈ L ∞ (0 , T ; H 3 (Ω) ∩ W 2 , ∞ (Ω)) required in the previous studies. Then, the global superconvergence result is deduced by interpolated post-processing approach. Finally, some numerical results are provided to verify the theoretical analysis. It should be mentioned that the analysis and results presented herein are also valid to some other known conforming and nonconforming finite elements. • The aim of this paper is to develop and investigate a fully-discrete scheme with conforming P 1 element for the nonlinear time-fractional Joule heating problem in which the Caputo derivative is approximated by the classical L 1 method. • A novel superclose estimate in the H 1 -norm is derived rigorously with some new analysis techniques under low regularity which improved the results in the previous studies. • The results obtained in our paper are also valid to conforming rectangular Q 1 element, nonconforming Q 1 r o t element on square meshes , and the nonconforming rectangular E Q 1 r o t element and so on. [ABSTRACT FROM AUTHOR]

The nonconforming modified Quasi-Wilson finite element method (FEM) is employed to investigate the unconditional superconvergence behavior of the convection–diffusion-reaction equation for the linearized energy stable Backward-Euler (BE) and the second order Backward differentiation formula (BDF2) fully discrete schemes on arbitrary quadrilateral meshes. Then, the energy stabilities of the discrete solutions for the proposed schemes are demonstrated through the mathematical induction, which is crucial to proving the unique solvabilities with Brouwer fixed point theorem. Based on the above achievements and the two typical characteristics of the element: one is that its consistency error in the broken H 1 -norm can be estimated as order O (h 2) when the exact solution belongs to H 3 (Ω) , which is one order higher than its interpolation error, and the other is that the nonconforming part of the function in the finite element space is orthogonal to the bilinear polynomials on each element of the subdivision in the integration sense, the superclose estimates without any restriction between spatial partition parameter h and the time step τ are obtained. Furthermore, the superconvergence properties for the aforementioned schemes are derived by the interpolation post-processing technique. Finally, the theoretical analysis is justified by the provided numerical experiments. • Superconvergence on quadrilateral meshes is got with modified Quasi-Wilson element. • The unique solvability of two schemes is proved by Brouwer fixed point theorem. • The reaction term is linearized and the approximation of the convective term is modified. • The inverse inequality is avoided to get unconditional superconvergence by stability. • Numerical examples in 2-D and 3-D space on rectangular and quadrilateral meshes are given. [ABSTRACT FROM AUTHOR]