两类典型多尺度方法的构造，分析和应用

We consider two classes of typical multiscale methods, atomistic/continuum methods for crystalline solids with defects and numerical homogenization methods for heterogeneous media. Those methods have many applications in materials science, biomedical sciences, and geophysics. The mathematical understanding of those methods is far from complete, and rigorous theory is absent in most physical relevant cases. The central mathematical question here, is to establish accurate and robust quantitative estimates for the consistency, stability, convergence rates and computational cost provided the dimension of original complex multiscale problems can be drastically reduced by those multiscale methods, and to construct multiscale methods which can achieve the optimal balance of the convergence rates and the computational cost based on rigorous mathematical justification. For atomistic/continuum coupling methods, we will carry out an intensive study on the coupling methods for point defects, and extend them to arbitrary crystal lattices in 3D and to line defect (dislocation). For numerical homogenization methods, we will deepen our understanding on linear problems, explore further for the large discrete heterogeneous network problem and nonlinear problems, and design optimal multiscale basis space. The in-depth investigation of those two classes of methods will lay a solid theoretical foundation for the study of multiscale problems, as well as for the design and application of mutliscale methods over a broader range of scales.

我们考虑两类典型的多尺度方法, 固体缺陷的原子/连续耦合方法和非均匀介质的数值均匀化方法。这两类方法在材料科学，生物医学，地球物理中有很多应用。在数学上，对这些方法的理解还很欠缺，大多数与物理问题相关的情形都没有严格的理论。这里的中心数学问题，是在计算方法大大降低原复杂多尺度问题自由度的基础上，对其相容性，稳定性，收敛性和计算复杂度等做出准确可靠的定量估计，并以此为依据设计出平衡计算量和精度的最优方法。对于原子/连续耦合方法，我们将对点缺陷问题深入研究，并将耦合方法推广到3维一般晶体和线缺陷(位错)，找到最佳的耦合界面位置和界面条件。对于数值均匀化方法，我们将在对线性问题深入研究的基础上，对大规模离散网络和非线性问题展开研究，设计最优的多尺度基函数空间。对这两类方法的深入研究将为在更广阔的尺度上研究多尺度问题，设计和应用多尺度方法打下坚实的理论基础。

在本项目中着重研究两类典型的多尺度方法, 即固体缺陷的原子/连续耦合方法和非均匀介质的数值均匀化方法。这里的中心数学问题，是在计算方法大大降低原复杂多尺度问题自由度的基础上，对其相容性，稳定性，收敛性和计算复杂度等做出准确可靠的定量估计，并以此为依据设计出平衡计算量和精度的最优方法。在本项目的资助下，我们结合相容性方法的准确性与混合型方法的易实现性，构造出在连续区域使用Cauchy-Born模型时具有最佳收敛阶数的BGFC方法，同时对原子/连续耦合方法的后验误差估计做了系统性的研究。基于数值均匀化方面的工作，构造了多尺度问题的快速算法，即Gamblet多分辨率方法，并延伸到区域分解，弹性力学方程等。对这两类方法的研究将为深入研究多尺度问题，设计和应用多尺度方法打下坚实的理论基础。