Fast Interior Penalty Methods

本项目将发展四阶及更高阶偏微分方程解的快速数值方法，使用内部惩罚方法。与使用协调、非协调或混合有限元的经典方法相比，内罚方法在计算复杂性、单元自然层次的存在、保持连续问题的对称正定性以及对于复杂问题易于导出收敛格式等方面具有优势。高阶问题的内罚方法的另一个显著优点是，高阶问题的间断有限元也适用于低阶问题。因此，内部惩罚方法的多重网格算法可以通过椭圆型问题的层级递归地发展。也就是说，二阶问题的多重网格算法可以自然地嵌入到四阶问题的多重网格算法中，然后再自然地嵌入到六阶问题的多重网格算法中，以此类推。这些多重网格法求解高阶问题的性能与求解二阶问题的多重网格法的性能相当。该项目将启动对高阶问题的内部惩罚方法的全面研究，以及多重网格、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解算器。该项目的结果将使解决一般区域上的六阶或更高阶问题成为可能。这些方法在应变梯度弹性、板屈曲、Monge-Ampere方程和Cahn-Hilliard方程中的应用也将被研究。本项目开发的快速算法将使科学家和工程师用高阶偏微分方程组模拟复杂现象成为现实。这些算法将增强不同领域的数值模拟的性能，如结构力学、流体力学、图像处理、纳米科学、几何光学、气象学、最优传输、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.