Finite Element Methods for Two Problems for Hyperbolic Partial Differential Equations

有限元法通常是确定偏微分方程近似解的首选方法，也是对其分析最彻底的方法。然而，在非线性双曲守恒律的设置中，例如，对于可压缩、无粘流和许多其他应用，情况并非如此。第一个项目的目标是为这种设置开发稳定、高效和准确的有限元方法。它涉及到分层有限元基础（hfeb）的使用。所使用的原理与用于定义谱粘度法的原理相似。目标是保持，至少远离不连续，有限元离散的全部精度，同时抑制不必要的和不稳定的振荡。与标准有限元基和光谱基相比，hfeb具有重要的优势。hfeb中固有的尺度分离（但标准碱基完全不存在）允许选择性地将人工粘度应用于离散溶液的高频成分。另一方面，根据最优网格级别定义的HFEB函数是本地支持的。这使得不连续点的位置易于识别（光谱基很难做到），这反过来又使网格细化策略的有效实施和人工扩散的空间选择性应用成为可能。所有这些理想的特征导致发展稳定，高精度双曲守恒定律的方法的可能性。将对多维双曲守恒定律进行分析和计算研究。其中需要解决的重要算法问题是如何最好地选择频率和空间相关的人工粘度系数，以及如何有效地实现高阶hfeb。第二个项目涉及双曲偏微分方程的精确控制性问题，在许多其他应用中，它影响振动结构的稳定和减少空气动力引起的噪声。一些初步的研究已经产生了有效的有限差分算法用于一维和几何上简单的二维波动方程问题。该算法是基于波动方程的离散化和初始和终端条件来产生和欠定线性系统。该系统作为一个优化问题的约束，该优化问题是为了提取可控性问题的特解而定义的。与以前的方法相比，新方法不需要正则化来获得收敛逼近。提出的工作包括将新的算法方法扩展到有限元设置，以便可以处理更复杂和现实的几何形状。还将考虑对变系数方程和双曲偏微分方程系统的扩展，例如线性弹性方程。算法研究将与基于优化方法的收敛特性分析相辅相成。拟议的项目与许多应用程序中的重要问题直接相关，因此它们的完成应该对如何解决这些问题产生重大影响。例如，尽管人们在开发超声速流动的计算技术方面已经付出了巨大的努力，但仍然需要开发能够用于一般几何形状（如飞机周围的流动）的高精度方法。第一个提出的项目正是这个目标，它的成功完成将直接影响解决航空航天、核、地球物理和其他领域问题的方式。此外，对于控制结构的不稳定运动或降低发动机产生的噪声的设置，尽管该设置与桥梁，飞机，建筑物，输电塔等设计问题直接相关，但现有的算法很少。因此，第二个项目的成功完成，其目标是开发、实施和分析高效、准确的计算方法，将对这些和其他应用产生重大影响。

Finite element methods are often the method of choice for determining approximate solutions of partial differential equations (PDEs) and they are usually also the most thoroughly analyzed. However, this is not the case in the setting of nonlinear hyperbolic conservation laws, e.g., for compressible, inviscid flows and many other applications. The goal of the first project is to develop stable, efficient, and accurate finite element methodologies for this setting. It involves the use of hierarchical finite element bases (HFEBs). The principles used are similar to those that serve to define spectral viscosity methods. The goal is to maintain, at least away from discontinuities, the full accuracy of the finite element discretization while simultaneously suppressing unwanted and unstable oscillations. HFEBs have important advantages over both standard finite element bases and spectral bases. The separation of scales inherent in HFEBs (but which is totally absent from standard bases) allows for the selective application of artificial viscosity to only the high frequency components of the discrete solution. On the other hand, HFEB functions defined with respect to the finest grid levels are locally supported. This leads to the easy identification of the positions of discontinuities (something difficult to do with spectral bases) which, in turn, enables the efficient implementation of grid refinement strategies and the spacially selective application of artificial diffusion. All of these desirable features lead to the possibility of developing stable, high-accuracy methods for hyperbolic conservation laws. Analytical and computational studies will be carried for multi-dimensional hyperbolic conservation laws. Among the important algorithmic questions that need to be answered are how to best choose the frequency and spacially-dependent artificial viscosity coefficient and how to efficiently implement higher-order HFEBs. The second project deals with exact controllability problems for hyperbolic PDEs that impact, among many other applications, the stabilization of vibrating structures and the reduction of aerodynamically induced noise. Some preliminary studies have resulted in efficient finite difference algorithms for one-dimensional and geometrically simple two-dimensional wave equation problems. The algorithms are based on discretizing the wave equation and the initial and terminal conditions to produce and underdetermined linear system. This system acts as a constraint for an optimization problem which is defined in order to extract particular solutions of the controllability problem. Compared to previous approaches, the new method does not require regularization to obtain convergent approximations. The proposed work includes the extension of the new algorithmic approach to the finite element setting so that more complicated and realistic geometries can be treated. Extensions to equations with variable coefficients and to systems of hyperbolic PDEs, e.g., the equations of linear elasticity, will also be considered. The algorithmic studies will be complemented with analyses of the convergence properties of the optimization-based methods.The proposed projects are directly related to important problems in many applications so that their completion should have significant impact on how those problems are solved. For example, although huge efforts have been devoted to developing computational techniques for supersonic flows, there is still a need for developing high-accuracy methods that can be implemented for general geometries such as flows about airplanes. The first proposed project has exactly this goal, and its successful completion would directly impact the way problems are solved in the aerospace, nuclear, geophysics, and other communities. Furthermore, for the setting of controlling the unstable motions of structures or for the reduction in noise produced by engines, there are few existing algorithms, despite the fact that that setting is directly related to problems in the design of bridges, airplanes, buildings, transmission towers, etc. Thus, the successful completion of the second project, which has as its goal the development, implementation, and analysis of efficient and accurate computational methods, would have great impact on these and other applications.