RUI: Investigation of Discontinuous Galerkin Least-Squares Finite Element Methods for Singularly Perturbed Problems

本计画的研究目的是探讨奇异摄动反应扩散问题的间断伽辽金最小二乘有限元方法。奇摄动问题的数值逼近是一个重要而又困难的课题。这类问题的解析解通常包含边界层或内层，这会导致标准有限元法和有限差分法解中的层附近出现非物理的过度数值振荡。许多稳定化技术已被开发，以改善数值解。这些技术的缺点是存在的问题依赖的参数，需要根据层的先验知识，这仍然是无法实现的，在最复杂的问题进行适当的调整。DG-LS有限元法是稳健和高效的，它不涉及这样的问题依赖的参数。本文的研究将有助于加深对DG-LS有限元和奇异摄动问题的认识和理解，为DG-LS有限元分析提供理论框架，并建立高效、高质量的自适应格式的工作原理。奇摄动问题在流体力学、电磁学、半导体研究、趋化性、遗传学和计算生物学等领域有着广泛的应用，因此引起了工程师、数学家和科学家的广泛关注。数值近似通常是解决这类问题的唯一方法。奇异摄动问题迫切需要一种与问题相关的无参数数值方法。该项目旨在开发创新的解决方案来满足这一需求，这将导致可靠和有效的数值方法来解决复杂的奇摄动反应扩散问题。

The research objective of this project is to investigate discontinuousGalerkin least-squares finite element methods (DG-LS FEMs) for reaction-diffusion problems with singular perturbations. The numerical approximation of singularly perturbed problems is a practically important but still difficult subject. The analytical solutions to such problems typically contain boundary or interior layers, which cause nonphysical excessive numerical oscillations in the vicinity of layers in the solutions by standard finite element method and finite difference method. Many stabilization techniques have been developed to improve numerical solutions. The drawback of these techniques is the presence of problem-dependent parameters that need to be properly tuned according to a priori knowledge of layers, which nonetheless is not achievable in most complex problems. The DG-LS FEM is robust and efficient, which does not involve such problem-dependent parameters. The proposed research will advance knowledge and understanding of the DG-LS FEM as well as singular perturbation problems, provide a theoretical framework for the analysis of DG-LS FEMs, and establish a working principle of efficient and high quality adaptive schemes. Singularly perturbed problems have attracted a lot of attention from engineers, mathematicians and scientists because of a wide range of important applications such as fluid dynamics, electromagnetism, semiconductor research, chemotaxis, genetics, and computational biology. Numerical approximations are usually the only way for solving such problems. There is an urgent need for a problem-dependent-parameter-free numerical method for singularly perturbed problems. This project is to develop creative solutions to fulfill this need, which will lead to reliable and efficient numerical approaches for solving complex singularly perturbed reaction-diffusion problems.