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

L1 中一阶偏微分方程的非线性有限元逼近

• 批准号: 0510650
• 负责人: Jean-Luc Guermond
• 金额: $ 67.75万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-15 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0510650&HistoricalAwards=false
• 关键词: NONLINEAR; FINITE; ELEMENT; APPROXIMATION; FIRST

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.

许多工程应用涉及偏微分方程，其中稳定性不是能量估计的结果。这是非线性守恒定律、平流主导流、多相流和自由边界问题的情况，在这些问题中，冲击锋和不连续面是驱动特征，给数值方法带来了重大困难。这些问题的自然稳定性设置包括可积性、有界变分或有界性。当人们想要保持像质量这样的量或者当人们想要保持像温度或密度这样的量的正性或有界性时，这种稳定性自然就会出现。研究人员提出了一种新的非线性逼近技术来求解上述一类微分方程。这种新方法包括在问题的自然稳定范数中计算最佳逼近，这是一种与标准技术完全不同的观点。研究人员将线性非最优视角（在能量空间中工作）转换为最优非线性视角（在类有界变分空间中工作）。尽管非线性算法更加复杂和难以分析，但它们在处理粗糙数据、复杂边界和刚性非线性时具有很大的优势。大量的工作已经在过去致力于发展鲁棒数值方法。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复杂过程的准确和忠实的数字表示。例如，模拟界面、冲击和尖锐前沿仍然是一个巨大的挑战。拟议的项目在许多领域具有广泛的影响。在机械和航空航天工程中，提出的方法改进了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相变的数值模型。在石油工程中，这套新方法有利于油藏多相流的模拟。总的来说，该项目还将对环境科学、地球物理学和纳米技术产生重大影响，这些领域需要强大的近似技术来解决冲击、尖锐界面和非线性现象。