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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]

Testing high-dimensional data independence is an essential task of multivariate data analysis in many fields. Typically, the quadratic and extreme value type statistics based on the Pearson correlation coefficient are designed to test dense and sparse alternatives for evaluating high-dimensional independence. However, the two existing popular test methods are sensitive to outliers and are invalid for heavy-tailed error distributions. To overcome these problems, two test statistics, a Spearman's footrule rank-based quadratic scheme and an extreme value type test for dense and sparse alternatives, are proposed, respectively. Under mild conditions, the large sample properties of the resulting test methods are established. Furthermore, the proposed two test statistics are proved to be asymptotically independent. The max-sum test based on Spearman's footrule statistic is developed by combining the proposed quadratic with extreme value statistics, and the asymptotic distribution of the resulting statistical test is established. The simulation results demonstrate that the proposed max-sum test performs well in empirical power and robustness, regardless of whether the data is sparse dependence or not. Finally, to illustrate the use of the proposed test method, two empirical examples of Leaf and Parkinson's disease datasets are provided. [ABSTRACT FROM AUTHOR]