Mathematical Sciences: Numerical Methods and Adaptivity for Nonlinear PDEs

该项目将支持对材料科学中出现的相变和自由边界问题的自适应有限元方法的设计、实施和分析方面的研究。将研究相变的数值方法；高梯度网格的使用对于相变和自由边界问题是极其重要的，这些问题通常表现出尖锐的界面和非常薄的过渡区。解或它的一些导数，可能在这些层内表现出不连续性或变化非常快，如果奇点得不到适当的解决，就会导致全局数值困难。还将研究显示表面张力和不稳定机制之间竞争的相关尖锐界面模型，例如固体晶体和凝固过程中的应力驱动不稳定性。研究强非线性偏微分方程线性化技术的设计，并与网格加密相结合。对流流体与自由边界的相互作用将被数值分析，例如对应于与其熔体共存的固体以及多相流体的界面。最后，将研究约束问题的混合方法，特别是Stokes流和相关的自由边界问题，以及板在障碍物上的自由边界问题。该项目涉及热力学、流体力学和弹性力学中的非线性偏微分方程组的自适应网格加密的设计和实现。对于这些问题中的许多问题，使用高梯度网格是很重要的，这些问题通常表现出尖锐的界面以及非常薄的过渡区域。人们对这类问题的兴趣与日俱增，不仅是因为它们的数学特性，而且还因为它们在材料科学中的相变、火焰传播、燃烧理论和晶体生长中的应用。

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.