Novel mixed and DG methods

对计算解技术的不断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模拟中越来越多地使用的两类方法是混合型有限元方法和间断Galerkin(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。本着这一精神，我们提出了以下五个方面的研究：(I)混合方法(Ii)间断Petrov-Galerkin(DPG)格式(Iii)杂交间断Galerkin(HDG)方法(Iv)光子膜的模拟和(V)复数轴对称模拟。第一部分讨论新的应力单元及其在具有弱施加应力对称的弹性力学混合方法中的意义。第二种是在方案设计中追求一种新的DPG范式，其中自动计算最优测试空间。第三，关于DG方法，它模仿混合方法，但具有灵活稳定的附加优势，并延续了以前由基金会支持的一系列研究。同时考虑了源问题和本征问题。剩下的两个研究方向考虑的是需要新的数学发展的应用，例如(Iv)需要良好的非线性特征解算器和(V)需要合理的奇点处理。这项拟议的研究带来了新的数学成分，催生了新的模拟方法。这些数学技巧的优点是适用范围广。因此，可以针对几个不同的应用领域，包括固体力学、传输现象、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，将新方法应用到流体流动中，可能会使飞机工业受益。可靠的模拟方法可以廉价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，将人力资源开发融入到活动中。

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.