Collaborative Research: Numerical Methods and Adaptive Algorithms for Sixth-Order Phase Field Models

这个项目将使用计算数学模型来加深对微乳液系统和晶体形成模型这两个应用的理解。第一类应用与油-水-表面活性剂体系有关，这些体系在石油开采、环境友好型溶剂的开发、消费者和商业清洁产品配方以及药物输送系统中非常重要。将要研究的晶体模型将有助于检测晶体材料中的拓扑缺陷，这是材料科学界非常感兴趣的任务。具体的例子包括过冷液体，延性材料中的裂纹扩展，以及与光子学和半导体、细胞结构衬底和磁共振造影剂相关的应用。 阻碍普通数学和科学界使用它们的一个主要挑战是缺乏对这些复杂系统的了解。这个 项目将建立高效的模拟算法，支持对这些过程的研究和先进材料的设计。该项目将为本科生和研究生提供机会，并向他们介绍最先进的数值方法的理论和实施。在这个项目中，PI将针对 的应用和数学模型两类开发C0内罚有限元方法。C0内罚有限元方法最初是用来处理力学中出现的四阶椭圆型问题的，但它已被应用于其他四阶和六阶偏微分方程组。本项目的重点是含时六阶偏微分方程组的数值方法。高阶导数与时间相关分量的结合给建立稳定、收敛和高效的数值方法来逼近这些模型的解带来了许多挑战。要完成的工作包括建立所提出的数值方法的唯一可解性、稳定性和收敛的形式证明。最大的挑战将是开发一个框架，以建立最优阶数误差估计。最后，为了提高所提出的数值方法的效率，PI计划利用算子分裂技术和基于目标导向的双重加权方法获得的后验误差估计的时空自适应来开发用于时空离散系统的高效求解器。该项目由计算数学计划和已建立的激励竞争研究计划(EPSCoR)联合资助。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project will use computational mathematics models to further the understanding of two applications, microemulsions systems and crystal formation models. The first class of applications has relevance for oil-water-surfactant systems, which are important in oil recovery, development of environmentally friendly solvents, consumer and commercial cleaning product formulations, and drug delivery systems. The crystal models to be studied will be useful in detecting topological defects within crystalline materials, a task which is of great interest in the material science community. Specific examples include supercooled liquids, crack propagation in a ductile material, and applications relating to photonics and semiconductors, cell structure substrates and MRI contrast agents. A major challenge impeding their use by the general mathematical and scientific community has been a lack of understanding of these complex systems. This project will build efficient algorithms for simulation that will support the study of these processes and the design of advanced materials. The project will provide opportunities to undergraduate and graduate students and introduce them to the theory and implementation of state-of-the-art numerical methods. In this project the PIs will develop C0 interior penalty finite element methods for the two classes of applications and mathematical models. The C0 interior penalty finite element method was originally constructed to handle fourth-order elliptic problems arising in mechanics, but its adaptations have been applied to other fourth- and sixth-order partial differential equations. The focus of this project is on numerical methods for time-dependent sixth-order partial differential equations. The high derivative order in combination with a time-dependent component presents many challenges to the creation of stable, convergent, and efficient numerical methods approximating solutions to these models. The work to be accomplished includes the establishment of formal proofs for the unique solvability, stability, and convergence of the proposed numerical methods. The largest challenge will be to develop a framework which establishes optimal order error estimates. Finally, in order to improve upon the efficiency of the proposed numerical methods, the PIs plan to develop efficient solvers for space-time discretized systems using operator-splitting techniques and space-time adaptivity based on a posteriori error estimates obtained by the goal-oriented dual weighted approach.This project is jointly funded by Computational Mathematics program, and by the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.