Fast Interior Penalty Methods

本项目将利用内罚方法开发四阶和高阶偏微分方程的快速数值方法。内罚方法在计算复杂度、元素自然层次的存在性、连续问题的对称正确定性的保持性、复杂问题的收敛格式的易于导出等方面，都优于经典的一致性、非一致性或混合有限元方法。高阶问题的内罚方法的另一个显著优点是，高阶问题的不连续有限元也适用于低阶问题。因此，内罚方法的多网格算法可以通过椭圆问题的层次递归发展。也就是说，二阶问题的多网格算法可以自然嵌入到四阶问题的多网格算法中，而四阶问题的多网格算法又可以自然嵌入到六阶问题的多网格算法中，以此类推。这些多网格方法对高阶问题的性能与多网格方法对二阶问题的性能相当。该项目将启动高阶问题的内部惩罚方法的综合研究，以及多网格、区域分解和自适应算法，这些算法将为所产生的离散问题提供快速求解器。本课题的研究结果将为在一般域上求解六阶及更高阶的问题提供可行性。这些方法在应变梯度弹性、板屈曲、monge - amere方程和Cahn-Hilliard方程中的应用也将被研究。在这个项目中开发的快速算法将使科学家和工程师能够通过高阶偏微分方程来模拟复杂现象。这些算法将提高数值模拟在不同领域的性能，如结构力学、流体力学、图像处理、纳米科学、几何光学、气象学、最佳运输、微分几何和晶体生长等。

This project will develop fast numerical methods for fourth and higher order partial differential equations using the interior penalty approach. The interior penalty approach has advantages over the classical approaches that use conforming, nonconforming or mixed finite elements in terms of the computational complexity, the existence of natuaral hierarchies of elements, the preservation of the symetric positive definiteness of the continuous problem, and the ease of deriving convergent schemes for complicated problems. Another significant advantage of interior penalty methods for higher order problems is due to the fact that discontinuous finite elements for higher order problems are also suitable for lower order problems. Therefore multigrid algorithms for interior penalty methods can be developed recursively through the hierarchy of elliptic problems. Namely, multigrid algorithms for second order problems can be embedded naturally in multigrid algorithms for fourth order problems, which can then be embedded naturally in multigrid algorithms for sixth order problems, and so on. The performance of these multigrid methods for higher order problems is comparable to the performance of multigrid methods for second order problems. This project will initiate a comprehensive study of interior penalty methods for higher order problems together with multigrid, domain decomposition and adaptive algorithms that will provide fast solvers for the resulting discrete problems. The results of this project will make it feasible to solve problems of order six and higher on general domains. Applications of these methods to strain gradient elasticity, plate buckling, the Monge-Ampere equations and the Cahn-Hilliard equations will also be investigated.The fast algorithms developed in this project will make it practical for scientists and engineers to model complex phenomena by higher order partial differential equations. These algorithms will enhance the performance of numerical simulations in diverse areas such as structural mechanics, fluid mechanics, image processing, nanoscience, geometric optics, meteorology, optimal transport, differential geometry, and crystal growth, among many others.