Discontinuous Galerkin Methods for PDEs with Heterogeneous Coefficients

异质性和各向异性是在许多不同的设置中遇到的现象：在多孔介质中的流动，光学层析成像，中子输运，辐射传输，电磁学等，而这些问题的微分方程建模可能在性质上是不同的，它们共享的共同特征是，它们涉及的参数可能是高度异质性（值可能有许多数量级的跳跃）或可以是高度各向异性的。在通常的术语下，这些问题是退化的（即没有一致的椭圆性），因此，不能使用标准的数学方法来处理。这个研究项目的目标是分析退化问题（适定性，稳定性等）。 并发展一般的间断Galerkin技术来逼近它们。将被构造的不连续伽辽金将是稳定的所有感兴趣的退化的情况下，将自动近似具有不同参数范围的子域之间的物理上有意义的传输条件，而无需用户必须识别具有特定属性的子域先验。关键是使用弗里德里希系统的数学框架，并设计适当的算子来控制近似解中的不连续性。所提出的近似技术的优点是，它从一个完全不同的角度来解决手头的问题，而不是通常的多域/多算法方法。 这项新技术将自动检测异质性/各向异性，并在用户无需手动操作的情况下适应各种情况。 这一研究项目的影响将是广泛的，因为所解决的问题涉及工程、环境科学、生物物理学、石油工程、半导体工业、提出一种新的鲁棒逼近技术来解决具有高度非均匀和各向异性特性的问题，最终将使控制或处理这类问题的许多科学和工程领域受益。仍然是一个严峻的挑战。

Heterogeneity and anisotropy are phenomena encountered in many different settings: flows in porous media, optical tomography, neutron transport, radiative transfer, electromagnetics, etc. While the differential equations modeling these problems may be different in nature, the common feature they share is that they involve parameters that may be highly heterogeneous (values may have jumps of many orders of magnitude) or can be highly anisotropic. Under the usual terminology these problems are degenerate (i.e. there is no uniform ellipticity) and, as a result, cannot be handled using standard mathematical methods. The objective of this research project is to analyze degenerate problems (well-posedness, stability, etc.) and to develop general discontinuous Galerkin techniques for approximating them. The discontinuous Galerkin that will be constructed will be stable for all the degenerate cases of interest and will automatically approximate the physically meaningful transmission conditions between sub-domains with different parameter ranges without the user having to identify sub-domains with specific properties a priori. The key is to use the mathematical framework of Friedrichs systems and to design the operators that controls discontinuities in the approximate solution appropriately.The merit of the proposed approximation technique is that it addresses the problem at hand from a radically different perspective than the usual multi-domain/multi-algorithmic approaches. The new technique will automatically detect heterogeneity/anisotropy and will adapt to situations without the user having to take manual action. The impact of this research project will be broad since the class of problems addressed touches many fields in engineering, in environmental sciences, in geophysics, in petroleum engineering, semiconductor industry, etc. Proposing a novel robust approximation technique for solving problems with highly heterogeneous and anisotropic properties will eventually benefit many areas of science and engineering where controlling or dealing with this type of problem is still a serious challenge.