Mathematical Sciences: Numerical Methods and Adaptivity for Nonlinear PDEs

该项目将支持设计方面的研究， 自适应有限元方法的实现和分析 对于相变和自由边界问题， 材料科学 相变的数值方法将 研究;使用高等级的网格是非常 重要的相变和自由边界问题， 典型地表现出尖锐的界面以及非常薄的 过渡区域。 解决方案，或其衍生物， 可能表现出不连续性或只是在这种不连续性内非常迅速地变化， 层，导致全球数值困难，如果 奇异性没有被正确地解决。 相关锐界面 模型表现出表面张力和 还将研究破坏稳定的机制，如压力 固体晶体和凝固中的驱动不稳定性 流程. 强线性化技术的设计 非线性偏微分方程将被研究， 结合网格细化。 对流的相互作用 具有自由边界的流体，例如对应于 固体与其熔体共存以及多相界面 流体，将进行数值分析。 最后，混合方法 将研究受约束的问题，特别强调 Stokes流动和相关的自由边界问题以及 障碍物上板的自由边界问题 本项目涉及的设计和实现 非线性偏微分的自适应网格细化 热力学、流体动力学和 弹性 使用高等级补片对于以下方面很重要： 这些问题中的许多，通常表现出尖锐的界面， 以及非常薄的过渡区域。 日益增长的兴趣 不仅源于它们的数学特征 也可以应用于材料的相变 科学、火焰传播、燃烧理论和晶体 增长

This project will support research in the design, implementation, and analysis of adaptive finite element methods for phase transitions and free boundary problems arising in material sciences. Numerical methods for phase transitions will be studied; the use of highly graded meshes is extremely important for phase transitions and free boundary problems which typically exhibit sharp interfaces as well as very thin transition regions. The solution, or some of its derivatives, may exhibit discontinuities or just vary very rapidly within such layers, leading to global numerical difficulties if the singularities are not properly resolved. Related sharp interface models exhibiting competition between surface tension and destabilizing mechanisms will also be studied, such as stress driven instabilities in solid crystals and solidification processes. The design of linearization techniques for strongly nonlinear partial differential equations will be studied and combined with mesh refinements. The interaction of convective fluid with free boundaries, for example that corresponding to solid coexisting with its melt and interfaces of multiphase fluids, will be analyzed numerically. Finally, mixed methods for constrained problems will be studied, with special emphasis on the Stokes flow and related free boundary problems as well as on free boundary problems for plates over obstacles. This project is concerned with the design and implementation of adaptive mesh refinements for nonlinear partial differential equations arising in thermodynamics, fluid dynamics, and elasticity. The use of highly graded meshes is important for many of these problems, which typically exhibit sharp interfaces as well as very thin transition regions. The increasing interest in such problems stems not only from their mathematical features but also their applications to phase transitions in material sciences, flame propagation, combustion theory, and crystal growth.