Hybridizable discontinuous Galerkin methods for higher order partial differential equations

为了理解、预测并最终控制一个复杂的自然或人造物理系统，人们经常使用偏微分方程对其进行建模。然而，在现代复杂应用中产生的几乎所有模型中，以基本数学函数的形式获得精确解是不可能的。因此，从业者必须求助于所谓的数值方法来计算定义模型的偏微分方程的近似解。因此，数值模拟在现代科学技术中发挥着关键作用。它们还允许在实际构建物理模型之前仅在计算机上设计和测试各种模型，从而大大降低制造成本。对实际问题的近似解的成功计算部分取决于计算机技术的进步。然而，更重要的是，它取决于设计、分析和实现高效、可靠、准确和稳健的数值方法。最广泛使用的数值方法之一是有限元方法，它已成为模拟科学和工程中出现的各种现象的不可或缺的工具，例如飞机，汽车，桥梁，石油平台以及最近的纳米材料的设计，仅举几例。不连续伽辽金（DG）方法构成了有限元方法的一个特殊亚族，以其稳定性、鲁棒性、通用性和高阶精度而闻名。在这个项目中，PI将开发和分析混合DG （HDG）方法来解决结构力学中出现的问题。特别强调的是设计这样的方法来处理诸如梁、板和壳等薄域的问题，因为它们提出的挑战在科学计算社区中引起了很大的兴趣。杂交过程允许消除许多全局耦合自由度，使线性系统明显小于其经典的DG对应系统。由此产生的HDG方法具有DG方法的理想特性，如稳定性、高阶收敛性和鲁棒性，并且在某些情况下它们表现出更好的特性。对这些现象的数学分析也是拟议项目的一部分。本项目由几个部分组成：Naghdi拱门的HDG方法；双调和的问题;Reissner-Mindlin板块;以及四阶时间相关问题。尽管每一个步骤本身都值得关注，但PI的最终目标之一是为壳模型设计有效的数值方法，而上述每一个步骤都是实现这一目标的踏脚石。

In order to understand, predict, and eventually control a complex natural or manufactured physical system, one often models it using partial differential equations. Nevertheless, in practically all models arising from complex modern applications, obtaining an exact solution in the form of basic mathematical functions is not a possibility. Thus, a practitioner must resort to what is called a numerical method for computing an approximate solution to the partial differential equations defining the model. Numerical simulations thus play a key role in modern science and technology. They also allow significant reduction in manufacturing costs by designing and testing various models merely on computers before actually building a physical model. Successful computation of approximate solutions to practical problems of interest in part depends on advances in computer technology. However, more importantly, it hinges upon the design, analysis, and implementation of efficient, reliable, accurate, and robust numerical methods.One of the most widely used family of numerical methods is the finite element method, which has become an indispensable tool for simulation of a wide variety of phenomena arising in science and engineering such as the design of aircrafts, automobiles, bridges, oil platforms, and more recently of nano-materials, to name a few. Discontinuous Galerkin (DG) methods constitute a special subfamily of finite element methods which are known for their stability, robustness, versatility, and high-order accuracy.In this project, the PI will develop and analyze hybridizable DG (HDG) methods for problems arising in structural mechanics. Particular emphasis will be on devising such methods for problems dealing with thin domains such as beams, plates, and shells, since they pose challenges which have attracted much interest in the scientific computing community. The hybridization procedure allows the elimination of many of the globally coupled degrees of freedom rendering the linear system significantly smaller than that of its classical DG counterparts. The resulting HDG methods enjoy desirable properties of DG methods such as stability, high-order convergence, and robustness, and in certain cases they exhibit even better properties. Mathematical analysis of such phenomena is also a part of the proposed project. This project consists of several parts: HDG methods for Naghdi arches; biharmonic problems; Reissner-Mindlin plates; and fourth-order time-dependent problems. Notwithstanding each of these steps is worthy of interest in its own right, one of the ultimate goals of the PI is to devise efficient numerical methods for shell models, and each one of the above steps constitute a stepping stone towards this goal.