Finite Element Methods for Higher Order Variational Inequalities

椭圆变分不等式是对涉及椭圆偏微分算子和约束优化的现象进行建模的基本数学工具。该项目将开发和分析四阶和高阶椭圆变分不等式的有限元方法，这些方法在力学和椭圆最优控制问题中自然出现。 PI 最近的一项理论进展表明，对于基尔霍夫板的位移障碍问题，误差分析的核心仅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。 这种新方法将扩展到其他具有不同类型约束的四阶及更高阶变分不等式，从而将成熟的边值问题有限元方法（一致和非一致方法、间断伽辽金方法、广义有限元方法、等参有限元方法、局部网格细化、奇异函数方法等）引入高阶变分不等式数值求解的研究中。还将开发高阶变分不等式的快速求解器，例如多重网格方法、域分解方法和自适应方法。 特别是，该项目将带来用于二阶椭圆分布式最优控制问题的新算法，其具有与现有算法根本不同的逐点状态和/或控制约束。 该项目的结果将为高阶变分不等式的数值求解提供新的见解，随着科学、工程和金融领域越来越复杂的现象通过高阶微分方程进行建模，这一领域变得越来越重要。该项目的成果将影响需要可靠且高效的数值算法来解决此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.