Mathematical Sciences: Finite Element Methods for Nonlinear PDE's and Adaptivity

小行星9623394 由非线性偏微分方程（PDE）控制的问题， 表现出急剧或扩散，但迅速，过渡是至关重要的， 科学与工程学液-固相变，多组分 合金，各向异性和结，外延生长，多相 不可压缩流体和平流为主的流动是相关的 本项目中考虑的多尺度示例。设计、分析和 实施创新的有限元方法， 自由边界、薄层和约束的存在，同时 通过自适应改进来科普可能的数值污染效应 mesh是这个项目的核心。先验稳定性和误差分析， 与后验误差控制一起， 提出的研究，揭示了有限的价值，线性理论在 强非线性偏微分方程。的后验误差估计 退化抛物型偏微分方程必须反映自由的独特存在， 边界和薄过渡层。相关PDE和计算问题 （实现，递归细化/粗化算法，多级 解算器）也将被研究。 该项目有望揭示高效可靠的自适应 材料科学中相变的有限元计算，以及 从而影响相变的高性能计算， 表面张力效应理解和计算非线性演化 在外延生长中观察到的波纹是至关重要的 在电子和材料科学方面。新的有限元方法将是 设计并用于探索应力驱动不稳定性的简单模型， 固体一个描述共存和相互作用的出租模型的验证 几种海洋物种对觅食、捕食者和环境的反应 条件在生态学和环境科学中至关重要。混合有限 元素方法将与指数拟合结合使用， 更好地利用非线性系统底层的结构趋序性， 平流和扩散两种极端情况下的偏微分方程主导了流动。

9623394 Nochetto Problems governed by nonlinear partial differential equations (PDEs) which exhibit sharp or diffuse, but rapid, transitions are of primary importance in science and engineering. Liquid-solid phase transitions, multicomponent alloys, anisotropy and junctions, epitaxial growth, multiphase incompressible fluids, and advection dominated flows are relevant multiscale examples considered in this project. The design, analysis and implementation of innovative finite element methods which handle the presence of free boundaries, thin layers and constraints, and the same time cope with possible numerical pollution effects via adaptively refined meshes is central to this project. A priori stability and error analyses, together with a posteriori error control are quite delicate aspects of the proposed research that reveal the limited value of linear theory in the context of strongly nonlinear PDEs. A posteriori error estimates for degenerate parabolic PDEs must reflect the distinctive presence of free boundaries and thin transition layers. Related PDE and computational issues (implementation, recursive refinement/coarsening algorithms, multilevel solvers) will also be investigated. This project is expected to shed light on the efficient and reliable adaptive finite element computation of phase transitions in materials science, and thus have impact on high performance computing of phase transitions with surface tension effects. Understanding and computing nonlinear evolutions of corrugations observed in epitaxial growth is of fundamental importance in electronics and materials science. Novel finite element methods will be designed and used to explore a simple model for stress driven instabilities in solids. Validation of a taxis model describing coexistence and interaction of several marine species in response to foraging, predators and environmental conditions is crucial in ecology and environmental sciences. Mixed finite element me thods will be used in conjunction with exponential fitting to better exploit the structure of taxis in the underlying system of nonlinear PDEs in both extremes of advection and diffusion dominated flows.