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.

变分不等式出现在涉及微分方程和优化的领域。 它们是对涉及不平等约束的科学、工程和金融现象建模的基本工具。 本项目的目标是设计，分析和实现可靠和有效的数值算法的变分不等式，涉及高阶微分方程，应用于最优控制和材料科学。新的有限元方法的高阶椭圆和抛物型变分不等式将开发与快速解决技术（自适应，并行和多级）的离散问题。重点是非光滑和非凸域的问题，其中的变分不等式的解决方案的规律性是更加微妙的，并为三维域的问题，数值计算是更苛刻的。 一个重要的应用是椭圆型偏微分方程约束的最优控制问题。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（经典协调有限元法、非协调有限元法、间断Galerkin方法、单位分解法、混合有限元法等）都得到了发展。和技术（误差估计，局部网格细化，包括奇异性的局部近似空间等）。可用于其数值解。 从这种方法设计的新的数值方法将从根本上不同于传统的方法，其中重点是控制变量。