四阶旋度电磁场方程协调谱元方法的设计、理论及应用

近年来，电磁逆散射理论与四阶磁流体动力学方程组等非常规电磁场计算的应用需求推动了基于四阶旋度算子的电磁场问题求解的国际前沿研究。本项目拟研究H(curl^2)-协调谱元。与H(curl)-协调谱元相比，H(curl^2)-协调谱元要求更强的连续性。与Lagrange型H(curl^2)-协调元相比，该协调谱元的基函数表达形式简单、具有分层结构，易于推广到高阶形式。H(curl^2)-协调谱元的构造不仅使得不高于四阶的旋度问题均有相应的协调谱元进行求解，并且能够有效地避免伪解。本项目预期设计一系列H(curl^2)-协调谱元，发展quad-curl问题的任意阶H(curl^2)-协调谱元方法，并分析其收敛性及超收敛性。最后将H(curl^2)-协调谱元方法首次应用于数值求解Maxwell传输特征值问题。项目成果不仅能完善谱元体系，更为相关的四阶电磁场问题的数值模拟提供了新途径。

Recently, there has been an increased interest in the fourth-order electromagnetic field problem (quad-curl problem) because of its important application in electromagnetic inverse scattering theory and magnetohydrodynamics equations. This project intends to study the H(curl^2)-conforming spectral element method. Compared with the H(curl)-conforming spectral element, the higher continuity requirement makes it challenging to construct H(curl^2)-conforming spectral element. Compared with the Lagrangrian H(curl^2)-conforming element, the basis functions of the conforming spectral element are simple and hierarchical so that they can be extended to higher-order form easily. The H(curl^2)-conforming spectral element can not only solve the quad-curl problems but also avoid spurious solutions. This project is expected to design H(curl^2)-conforming spectral elements on triangles, tetrahedrons, triangular prisms, tetragons, and hexahedrons, develop H(curl^2)-conforming spectral element method of arbitrary order for quad-curl problem, and analyze its convergence and superconvergence. Finally, the H(curl^2)-conforming spectral element method is firstly applied to solve the Maxwell transmission eigenvalue problem. The project will provide a new idea for the numerical simulations of the fourth-order electromagnetic field problems. At the same time, the results of this project can be applied to solve other practical problems involving the fourth-order curl operators.