NONLINEAR FINITE ELEMENT APPROXIMATION OF FIRST-ORDER PDE'S IN L1

许多工程应用涉及偏微分方程组，其中稳定性不是能量估计的结果。这对于非线性守恒律、平流为主的流动、多相流和自由边界问题都是如此，其中激波锋面和不连续是驱动特征，并给数值方法带来很大的困难。这些问题的自然稳定性设置涉及可积性、有界变差或有界性。当一个人想要保持像质量这样的量，或者当他想保持像温度或密度这样的量的正性或有界性时，这种稳定性自然发生。研究人员建议开发一种新的非线性逼近技术来求解上述类型的微分方程。这种新的方法包括计算问题的自然稳定范数的最佳逼近，这与标准方法的观点是完全不同的。研究人员从线性的非最优视角(在能量空间中工作)换取了最优的非线性视角(在有界变化的类空间中工作)。尽管非线性算法比较复杂和难以分析，但它们在处理粗糙数据、复杂边界和强非线性时具有很大的优势。过去，人们致力于发展稳健的数值方法。在一些领域已经取得了重大进展，但目前的技术水平远远不能提供对复杂过程的准确和忠实的数字表示。例如，模拟界面、冲击和锋面仍然是一个巨大的挑战。拟议中的项目在许多领域产生了广泛的影响。在机械和航天工程中，提出的方法改进了数值模型，用于模拟高速气体动力学、非线性弹性问题和形状记忆合金等新材料的相变。在石油工程中，这套新的方法有利于模拟油藏中的多相流。总体而言，该项目还将对环境科学、地球物理和纳米技术产生重大影响，在这些领域，需要解决冲击、尖锐界面和非线性现象的稳健近似技术。

Many engineering applications involve partial differential equations where stability is not the result of an energy estimate. This is the case for nonlinear conservation laws, advection-dominated flows, multi-phase flows, and free-boundary problems, where shocks fronts and discontinuities are driving features and pose significant difficulties for numerical methods. The natural stability setting for these problems involves integrability, bounded variations, or boundedness. This kind of stability naturally occurs when one wants to preserve quantities like mass or when one wishes to preserve the positivity or the boundedness of quantities like temperature or density. The investigators propose to develop a new nonlinear approximation technique for solving the above class of differential equations. This new approach consists of computing the best approximation in the natural stability norm of the problem, which is a radically different point of view than that of standard techniques. The investigators trade a linear non-optimal perspective (working in energy spaces) for an optimal nonlinear one (working in bounded-variation-like spaces). Even though the nonlinear algorithms are more complicated and difficult to analyze, they yield great benefits when working with rough data, complicated boundary, and stiff nonlinearities.A large amount of work has been dedicated in the past to the development of robust numerical methods. Significant progress has been made in some areas, but the current state of the art is far from providing accurate and faithful numerical representations of complex processes. For instance, simulating interfaces, shocks, and sharp fronts is still an enormous challenge. The proposed project has a broad impact in many fields. In mechanical and aerospace engineering, the proposed method improves numerical models for simulating high velocity gas dynamics, nonlinear elasticity problems, and phase transition in new materials like shape memory alloys. In petroleum engineering the new set of methods is beneficial for simulating multi-phase flows in reservoirs. In general, the project will also have significant impact on environmental sciences, geophysics, and nanotechnologies where robust approximation techniques for solving shocks, sharp interfaces, and nonlinear phenomena are needed.