Lixiu Wang

Qian Zhang, Lixiu Wang, Zhimin Zhang

In this paper, we first construct the $H^2$(curl)-conforming finite elements both on a rectangle and a triangle. They possess some fascinating properties which have been proven by a rigorous theoretical analysis. Then we apply the elements to construct a finite element space for discretizing quad-curl problems. Convergence orders $O(h^k)$ in the $H$(curl) norm and $O(h^{k-1})$ in the $H^2$(curl) norm are established. Numerical experiments are provided to confirm our theoretical results.

Changhui Yao, Yanping Lin, Lixiu Wang et al.

In this paper, the magneto-heating coupling model is studied in details, with turbulent convection zone and the flow field involved. Our main work is to analyze the well-posed property of this model with the regularity techniques. For the magnetic field, we consider the space $H_0(curl)\cap H(div_0)$ and for the heat equation, we consider the space $H_0^1(Ω)$. Then we present the weak formulation of the coupled magneto-heating model and establish the regularity problem. Using Roth's method, monotone theories of nonlinear operator, weak convergence theories, we prove that the limits of the solutions from Roth's method converge to the solutions of the regularity problem with proper initial data. With the help of the spacial regularity technique, we derive the results of the well-posedness of the original problems when the regular parameter $ε\longrightarrow 0$. Moreover, with additional regularity assumption for both the magnetic field and temperature variable, we prove the uniqueness of the solutions.