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

由非线性偏微分方程（PDEs）控制的Nochetto问题在科学和工程中具有重要意义，这些问题表现为急剧或扩散但迅速的过渡。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主导流动是本项目考虑的相关多尺度例子。设计、分析和实施创新的有限元方法，处理自由边界、薄层和约束的存在，同时通过自适应细化网格处理可能的数值污染效应，是本项目的核心。先验稳定性和误差分析以及后验误差控制是所提出的研究中相当微妙的方面，揭示了线性理论在强非线性偏微分方程中的有限价值。退化抛物型偏微分方程的后验误差估计必须反映自由边界和薄过渡层的独特存在。相关的PDE和计算问题（实现，递归细化/粗化算法，多层次求解器）也将被调查。本课题可望对材料科学中高效可靠的相变自适应有限元计算有所启发，进而对具有表面张力效应的相变的高性能计算产生影响。理解和计算外延生长中观察到的波纹的非线性演化在电子和材料科学中具有重要的基础意义。新的有限元方法将被设计和用于探索固体应力驱动不稳定的简单模型。在生态和环境科学中，验证描述几种海洋物种对觅食、捕食者和环境条件的共存和相互作用的计程车模型是至关重要的。混合有限元方法将与指数拟合结合使用，以更好地利用在平流和扩散主导流动的两个极端下非线性偏微分方程底层系统中的趋向性结构。

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.