Finite Element Methods for Higher Order Variational Inequalities

椭圆型变分不等式是模拟涉及椭圆偏微分算子和约束最优化现象的基本数学工具。这个项目将开发和分析四阶和更高阶椭圆型变分不等式的有限元方法，这是在力学和椭圆最优控制问题中自然产生的。PI的最新理论进展表明，对于Kirchhoff板的位移障碍问题，误差分析的核心只涉及连续级别的问题，因此任何适用于四阶边值问题的有限元方法也可以适用于障碍问题。将这一新方法推广到其他具有不同约束类型的四阶或更高阶变分不等式，将为边值问题(协调和非协调方法、间断Galerkin方法、广义有限元方法、等参有限元方法、局部网格加密、奇异函数方法等)带来成熟的有限元方法。研究了高阶变分不等式的数值解。还将开发高阶变分不等式的快速求解器，如多重网格法、区域分解法和自适应方法。特别是，这个项目将为二阶椭圆分布式最优控制问题带来新的算法，这些问题具有逐点状态和/或控制约束，与现有算法有根本不同。这个项目的结果将为高阶变分不等式的数值解提供新的见解，随着科学、工程和金融中越来越多的复杂现象被高阶微分方程建模，这个领域变得越来越重要。该项目的结果将影响到需要可靠和有效的数值算法来解决这类不平等问题的各个领域。

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.