Structure-Preserving Numerical Methods for Strongly Nonlinear Elliptic Partial Differential Equations

该研究项目的目标是开发准确有效的数值方法来求解强非线性椭圆偏微分方程（PDE）。 非线性问题在科学和工程中普遍存在，自然产生于材料科学、非线性弹性、流体动力学和图像处理。该提案研究了保留非线性偏微分方程基本属性的有效数值算法的分析和设计。该项目将为液晶材料和形状设计的研究人员提供高效的数值算法。除了设计高效算法外，该项目还将分析方法的稳定性和收敛率。与大量的线性问题文献相比，强非线性椭圆问题数值分析的工作相对较少。该项目的成功将为研究非线性现象的数值方法的未来发展提供启示。该项目分为三个不同的部分，即向列液晶的Landau-De Gennes模型的数值近似、蒙日安培偏微分方程的数值近似、数值最优传输问题。该项目的具体目标包括（i）构建基于分段线性或节点函数的新型数值方法，以保留离散最大原理，这是这些问题的基本属性，（ii）将稳健的低阶方法与精确的高阶方法相结合，并开发后验误差估计和自适应性，以提高方法的准确性和效率，（iii）基于非线性偏微分方程工具的离散版本对这些方法进行分析，例如伽马收敛和离散亚历山德罗夫最大原理，（iv）将这些方法应用于模拟液晶材料和天线该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The goal of this research project is to develop accurate and efficient numerical methods to solve strongly nonlinear elliptic partial differential equations (PDEs). Nonlinear problems are ubiquitous in science and engineering and arise naturally from materials science, nonlinear elasticity, fluid dynamics and image processing. This proposal studies the analysis and design of efficient numerical algorithms which preserve essential properties of nonlinear PDEs. The project will provide efficient numerical algorithms for researchers in liquid crystal materials and shape design. In addition to the design of efficient algorithms, the project will also analyze the stability and rates of convergence of the methods. Compared with the vast literature on linear problems, the work on numerical analysis for strongly nonlinear elliptic problems are relatively few. The success of the project will provide insight in future development of numerical methods in studying nonlinear phenomena. The project splits into three different parts, namely, numerical approximation of the Landau-De Gennes model of nematic liquid crystals, numerical approximation of the Monge Ampere PDEs, numerical optimal transportation problem. The specific goal of the project includes (i) construction of novel numerical methods based on piecewise linear or nodewise functions to preserve discrete maximum principle, an essential property of these problems, (ii) combination of robust lower order method with accurate higher order methods and development of a posteriori error estimation and adaptivity to improve the accuracy and efficiency of the methods, (iii) analysis of these methods based on discrete version of nonlinear PDE tools, such as Gamma convergence and discrete Alexandroff maximum principle, (iv) applying these methods in simulating liquid crystal materials and antenna design.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.