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

我们得到了几个结果，这些结果与以下代表性界面现象有关：“Belousov-Zhabotinsky反应”，“粘性指指”和“吸收介质中的支撑行为”。1)发展了“奇异极限法”、“分岔理论”、“固有粘度解”和“水平集法”，使我们能够分析这种现象出现的机理。例如，行波和螺旋波的出现，以及描述晶体外延生长中观察到的一些聚束现象的模型方程解的整体存在性。2)构造了一种基于奇异极限法的差分格式，使我们能够数值计算三维Stefan问题。3)指纹的生长可以通过考虑流变特性的非线性来解释；也就是说，建立数学模型变得容易了。(4)建立了一种模拟吸波介质中支撑物流动行为的数值方法，并通过数学分析证明了支撑物分裂和不分裂现象的合理性。5)构造了一种用多精度算法跟踪水平集的数值方法，可以捕获行波解。6)构造了带误差估计的有限元方法和域分解方法，实现了无法直接观测到的脑脊液和地幔对流的动力学行为。在此基础上，提出了一种利用无网格虚拟节点沿域界面进行并行有限元分析的高性能域分解方法。

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.