Development of Numerical Methods for Dynamics of Interfaces and its Applications to Experiments in Science and Engineering

我们得到了几个有关界面的典型现象：“Belousov-Zhabotinsky反应”、“粘性指进”和“吸附介质中的支撑行为”的结果。1）发展了“奇异极限方法”、“分歧理论”、“固有粘性解”和“水平集方法“，使我们能够分析这些现象出现的机理。例如，行波和螺旋波的出现，2）构造了一个基于奇异极限方法的差分格式，并对三维Stefan问题进行了数值计算; 3）在考虑晶体生长过程中的非线性效应的情况下，可以解释指状结构的生长，并给出了数值结果。流变性能的线性; 4）建立了一种数值方法，实现了通过吸收介质的流动的支撑行为，并通过数学分析证明了支撑分裂和非分裂现象的合理性。5）建立了一种多精度算法跟踪水平集的数值方法，6）构造了一种带误差估计的有限元方法和一种区域分解方法，使我们能够实现无法直接观测的脑脊液和地幔对流的动力学行为。在此基础上，提出了一种基于无网格虚拟节点的高性能区域分解并行有限元方法。

We obtained several results, which are concerned with the following representative phenomena of inter-faces : "Belousov-Zhabotinsky reaction", "Viscous fingering" and "Behavior of support in an absorbingmedium".1)"Singular limit method", "Bifurcation theory", "Proper viscosity solutions" and "Level set method"are developed and enable us to analyze the mechanism of the appearance of such phenomena. For example, the appearance of a travelling wave and a helical wave, and the global existence of the solution of the model equation describing some bunching phenomena observed in the epitaxial growth of crystals.2)A difference scheme based on the singular limit method is constructed and enables us to compute the 3-dimensional Stefan problem numerically.3)The growth of finger patterns can be interpreted by taking into account the non-linearity in the rheological properties ; that is, it becomes easy to construct the mathematical model.4)A numerical method which realizes the behavior of the support of the flow through an absorbing medium is developed, and the support splitting and non-splitting phenomena are justified by the mathematical analysis.5)A numerical method with the multiple precision arithmetic for tracking the level set is constructed, and can capture the travelling wave solutions.6)A finite element method with error estimates and a domain decomposition method are constructed, and enable us to realize the dynamical behavior of the cerebrospinal fluid and the Earth's mantle convection, which are unable to be observed directly. Furthermore, a high-performance domain decomposition method is proposed for the parallel finite element analysis using meshless virtual nodes along the domain interface.