Discontinuous Galerkin Methods for PDEs with Heterogeneous Coefficients

非均质性和各向异性是在许多不同的环境中遇到的现象：多孔介质中的流动、光学层析成像、中子输运、辐射传递、电磁等。虽然模拟这些问题的微分方程在性质上可能不同，但它们的共同特征是它们涉及的参数可能是高度异构的（值可能有许多数量级的跳跃），或者可能是高度各向异性的。在通常的术语下，这些问题是退化的（即没有一致的椭圆性），因此不能用标准的数学方法来处理。本研究项目的目的是分析退化问题（适定性，稳定性等），并开发一般的不连续伽辽金技术来逼近它们。所构建的不连续伽辽金对于所有感兴趣的退化情况都是稳定的，并且可以自动逼近具有不同参数范围的子域之间物理上有意义的传输条件，而无需用户先验地识别具有特定属性的子域。关键是使用弗里德里希系统的数学框架，并设计适当控制近似解不连续的算子。所提出的近似技术的优点在于，它从一个与通常的多域/多算法方法完全不同的角度来解决手头的问题。新技术将自动检测非均质性/各向异性，无需用户手动操作即可适应各种情况。这个研究项目的影响将是广泛的，因为所涉及的问题涉及工程、环境科学、地球物理、石油工程、半导体工业等许多领域。提出一种新的鲁棒近似技术来解决具有高度异质和各向异性性质的问题，最终将使许多科学和工程领域受益，在这些领域控制或处理这类问题仍然是一个严重的挑战。

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.