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.

变分不平等出现在涉及微分方程和优化的区域中。 它们是涉及不平等约束的科学，工程和金融中现象建模的基本工具。 该项目的目的是针对涉及高阶差异方程式的变异不平等设计，分析和实施可靠，有效的数值算法，并应用于最佳控制和材料科学。将有限的元素方法用于高阶椭圆形和抛物线型不平等，将与快速解决方案技术一起开发出来的问题（适应性裁缝）（适应性级别）（并稳定）（并稳定）（并稳定）。重点是在非平滑和非convex域上的问题，在这些域上的定期变化不等式的规律性更为微妙，并且在数值计算要求更高的三维域上的问题。 一个重要的应用是最佳控制问题受椭圆偏微分方程限制的最佳控制问题。通过将这些最佳控制问题重新调整为状态可变的第四阶变异不平等，各种有限元方法（经典的构象和不合格的有限元方法，不连续的盖勒金方法，统一方法的分配，混合有限元方法的分配，混合有限元方法等）和技术范围（局部网状范围，近距离精炼范围，近距离范围，近距离范围，遍布局部范围，近距离的范围，包括单独的范围，包括单独的范围，遍历了单独的范围，遍历了单独的范围。解决方案。 从这种方法设计的新数值方法将与传统方法所获得的方法从根本上不同，而传统方法的重点是控制变量。