Finite Element Methods for Higher Order Variational Inequalities

高阶变分不等式的有限元方法

• 批准号: 1319172
• 负责人: Susanne Brenner
• 金额: $ 24.48万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-15 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1319172&HistoricalAwards=false
• 关键词: Finite; Element; Methods; Higher; Order

Elliptic variational inequalities are fundamental mathematical tools for modeling phenomena that involve elliptic partial differential operators and constrained optimization. This project will develop and analyze finite element methods for fourth and higher order elliptic variational inequalities, which arise naturally for example in mechanics and elliptic optimal control problems. A recent theoretical advance by the PIs demonstrates that, for the displacement obstacle problems of Kirchhoff plates, the heart of the error analysis involves only problems at the continuous level and therefore any finite element method that works for fourth order boundary value problems can also be adapted for obstacle problems. This new approach will be extended to other fourth and higher order variational inequalities with different types of constraints, which will bring well-developed finite element methodologies for boundary value problems (conforming and nonconforming methods, discontinuous Galerkin methods, generalized finite element methods, isoparametric finite element methods, local mesh refinement, singular function method, etc.) into the study of numerical solution of higher order variational inequalities. Fast solvers for higher order variational inequalities, such as multigrid methods, domain decomposition methods and adaptive methods, will also be developed. In particular this project will lead to new algorithms for second order elliptic distributed optimal control problems with pointwise state and/or control constraints that are fundamentally different from existing algorithms. The results from this project will provide new insights to the numerical solution of higher order variational inequalities, an area that is becoming increasingly important as more and more complex phenomena in science, engineering and finance are being modeled by higher order differential equations. The outcomes of this project will impact diverse areas that require reliable and efficient numerical algorithms for the solution of such inequalities.

椭圆形的变分不平等是用于建模现象的基本数学工具，涉及椭圆形偏差算子和约束优化的现象。该项目将开发和分析针对第四和高阶椭圆形不平等现象的有限元方法，例如在力学和椭圆形最佳控制问题中自然出现。 PI的最新理论进步表明，对于Kirchhoff板的位移障碍物问题，错误分析的核心仅涉及在连续级别上的问题，因此，对于第四阶边界价值问题有效的任何有限元方法也可以适应障碍问题。 This new approach will be extended to other fourth and higher order variational inequalities with different types of constraints, which will bring well-developed finite element methodologies for boundary value problems (conforming and nonconforming methods, discontinuous Galerkin methods, generalized finite element methods, isoparametric finite element methods, local mesh refinement, singular function method, etc.) into the study of numerical solution of higher order variational不平等。还将开发出高阶变量不平等的快速求解器，例如多机方法，域分解方法和自适应方法。 特别是，该项目将导致二阶分布式分布式最佳控制问题的新算法，其最佳状态和/或控制约束与现有算法根本不同。 该项目的结果将为高阶变异不平等的数值解决方案提供新的见解，这一领域越来越重要，因为科学，工程和金融中越来越复杂的现象正在以高阶微分方程为模型。该项目的结果将影响需要可靠，有效的数值算法来解决此类不平等现象的不同领域。