复杂形状超导材料的高效算法研究

Superconducting material has very important applications in defense and economics, and the effective numerical method plays an irreplaceable role in studying the nature of superconducting material. In view of the needs of practical problems, there exits an urgent requirement of the efficient and high resolution numerical methods. Not only the approximation accuracy of the algorithm needs to be continuously improved, but the applicability also needs to be expanded, which requires that the algorithm is suitable to simulate superconducting material with a variety of complex shapes. In order to meet the needs of the actual computation, this project is expected to make some innovative work in efficient algorithm for simulating superconducting material. The project will develop a new efficient algorithm using the finite element exterior calculus theory, which effectively simulates the change of superconducting material in the applied magnetic field. The new algorithm improves the accuracy of the approximations for multiple physical quantities, and achieves the optimal accuracy allowed by the discrete spaces. Moreover, the algorithm is also suitable to simulate a variety of high-dimensional superconducting material having shapes with complex geometric and topological properties.

超导材料在国防事业和国民经济中有着极其重要的应用，高效的数值方法对于研究超导材料的性质起着不可替代的作用。由于实际问题的需要，对数值计算的要求变得越来越高，不仅计算精度要不断提高，而且算法的适用性也要不断扩大，要求能够满足对各种复杂形状超导材料进行数值模拟的需要。为满足这种实际计算的最新需求，本项目预期在超导模拟的高效算法方面做出一些创新性工作。本项目将致力于利用有限元外微分理论发展一种新的高效算法，能够有效模拟超导材料在外加磁场作用下的变化情况，并改进多个物理量的逼近精度，使数值解达到离散空间容许的最优阶精度。而且，该算法同样适用于数值模拟各种高维、具有复杂几何、拓扑形状的超导材料。

超导材料在国防事业和国民经济中有着极其重要的应用，但是高温强磁会破坏超导体的超导性质，因此有效的数值模拟对于研究超导材料的性质起着不可替代的作用。另外，由于实际问题的需要，对数值计算的要求变得越来越高，不仅计算精度要不断提高，而且算法的适用性也要不断扩大，要求能够满足对各种不规则超导体进行数值模拟的需要。本项目从线性化模型入手，利用有限元外微分理论构造一种有效的有限元算法，使之适用于各种拓扑不平凡、非凸的区域。而且，该算法还推广应用来数值模拟具有复杂形状的超导体在外加磁场作用下的变化情况，改进多个物理量的逼近精度，使数值解达到离散空间容许的最优阶精度。此外，还构造了新的导数恢复型有限元算法和多重网格算法，用来求解某些非线性问题，这些方法在保持精度的同时大大节省了计算量。因此，这些研究不仅在理论分析方法上带来创新，对实际应用也有一定价值。