Higher Order Variational Inequalities: Novel Finite Element Methods and Fast Solvers

高阶变分不等式：新颖的有限元方法和快速求解器

• 批准号: 1620273
• 负责人: Susanne Brenner
• 金额: $ 35.68万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620273&HistoricalAwards=false
• 关键词: Higher; Order; Variational; Inequalities; Novel

Variational inequalities appear in areas that involve differential equations and optimization. They are fundamental tools for the modeling of phenomena in science, engineering and finance that involve inequality constraints. The goal of this project is to design, analyze and implement reliable and efficient numerical algorithms for variational inequalities that involve higher order differential equations, with applications to optimal control and materials science.Novel finite element methods for higher order elliptic and parabolic variational inequalities will be developed together with fast solution techniques (adaptive, parallel and multilevel) for the resulting discrete problems. The emphasis is for problems on nonsmooth and nonconvex domains where the regularity of the solutions of the variational inequalities is much more subtle, and for problems on three dimensional domains where numerical computations are much more demanding. An important application is to optimal control problems constrained by elliptic partial differential equations. By reformulating these optimal control problems as fourth order variational inequalities for the state variable, various finite element methodologies(classical conforming and nonconforming finite element methods, discontinuous Galerkin methods, partition of unity methods, mixed finite element methods, etc.) and techniques (error estimators, local mesh refinements, inclusion of singularities in local approximation spaces, etc.) can be employed in their numerical solutions. The new numerical methods designed from this approach will be fundamentally different from the ones obtained by the traditional approach where the emphasis is on the control variable.

变分不等出现在涉及微分方程和最优化的领域。它们是对科学、工程和金融中涉及不平等约束的现象进行建模的基本工具。这个项目的目标是设计、分析和实现可靠和高效的数值算法来解决涉及高阶微分方程的变分不等式问题，并将其应用到最优控制和材料科学中。高阶椭圆型和抛物型变分不等式的新的有限元方法将与所产生的离散问题的快速求解技术(自适应、并行和多层)一起发展。重点是关于非光滑和非凸域上的问题，其中变分不等式解的正则性要微妙得多，以及对于三维区域上的问题，其中数值计算要求更高。它的一个重要应用是研究椭圆型偏微分方程约束下的最优控制问题。通过将这些最优控制问题转化为状态变量的四阶变分不等式，各种有限元方法(经典协调和非协调有限元方法、间断Galerkin方法、单位分解法、混合有限元方法等)和技术(误差估计器、局部网格细化、在局部近似空间中包含奇点等)。可以用于他们的数值解中。由这种方法设计的新的数值方法将从根本上不同于传统方法所得到的侧重于控制变量的方法。