Novel mixed and DG methods

对计算解决技术日益增长的需求要求开发新的和改进的方法。在物理和工程系统的仿真中，越来越多地使用两类方法：混合型有限元方法和不连续伽辽金（DG）型有限元方法。为这些方法建立良好的数学基础可以提高它们的可靠性，揭示改进它们的途径，并有助于发现全新的方法。本着这种精神，提出了以下五个研究方向：(i)混合方法；（ii）不连续Petrov-Galerkin （DPG）方案；（iii）杂交不连续Galerkin （HDG）方法；（iv）光子膜的模拟；(v)复杂轴对称模拟。第一部分讨论了新的应力单元及其在弱应力对称弹性混合方法中的意义。第二步是在自动计算最优测试空间的方案设计中追求一种新的DPG范式。第三，关于DG方法，它模仿混合方法，但具有灵活稳定的额外优势，并继续以前由基金会支持的一系列研究。同时考虑了源问题和特征问题。剩下的两条研究线，考虑了需要新的数学发展的应用，例如，（iv）需要好的非线性特征解算器和(v)需要对奇点的合理处理。计算机仿真方法是现代科学研究中不可缺少的工具。提出的研究带来了新的数学成分，催生了新的模拟方法。这些数学技术具有广泛适用的优点。因此，可以针对几个不同的应用领域，包括固体力学、输运现象、流体流动、波传播、触发闪电和纳米光子膜。举几个详细的例子，通过工业和学术合作，将新方法应用于流体流动，可能会使飞机工业受益。可靠的模拟方法可以低成本地指导下一代纳米光子器件的实验。最后，通过培训和研究生参与研究，将人力资源开发纳入活动。

Ever increasing demands on computational solution techniques necessitate development of new and improved methods. Two classes of methods, increasingly being used in simulation of physical and engineering systems, are finite element methods of the mixed type and the discontinuous Galerkin (DG) type. Building sound mathematical foundations for these methods increases their reliability, reveals avenues to improve them, and helps discover radically new methods. In this spirit, five lines of research are proposed on the following topics: (i) mixed methods (ii) discontinuous Petrov-Galerkin (DPG) schemes(iii) hybridizable discontinuous Galerkin (HDG) methods (iv) simulation of photonic membranes, and (v) complex axisymmetric simulations. The first deals with new stress elements and their implications in mixed methods for elasticity with weakly imposed stress symmetry. The second pursues a new DPG paradigm in the design of schemes where optimal test spaces are automatically computed. The third, concerns DG methods that mimic mixed methods, yet having the added advantage of flexible stabilization, and continues a line of research previously supported by the foundation. Both source problems and eigenproblems are considered. The remaining two lines of research, considers applications in need of new mathematical developments, e.g., (iv) needs good nonlinear eigensolvers and (v) needs sound treatment of singularities.Methods for computer simulation are an indispensable tool in modern scientific research. The proposed research brings fresh mathematical ingredients that spawn novel simulation methods. These mathematical techniques have the advantage of being broadly applicable. Accordingly, several disparate application areas can be targeted, including solid mechanics, transport phenomena, fluid flow, wave propagation, triggered lightning, and nanophotonic membranes. To detail a few examples, application of the new methods to fluid flow, through industrial and academic collaborations, can potentially benefit the aircraft industry. Reliable simulation methods can inexpensively guide experimentation of next generation nanophotonic devices. Finally, human resource development is integrated into the activities through training and participation of graduate students in the research.