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

本研究项目的目标是发展精确和有效的数值方法来求解强非线性椭圆型偏微分方程（PDEs）。非线性问题在科学和工程中无处不在，并且自然地从材料科学、非线性弹性、流体动力学和图像处理中产生。本文研究了保持非线性偏微分方程基本性质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有效的数值算法。除了设计高效的算法外，该项目还将分析这些方法的稳定性和收敛速度。与线性问题的大量文献相比，强非线性椭圆型问题的数值分析工作相对较少。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见解。该项目分为三个不同的部分，即向列液晶的Landau-De Gennes模型的数值逼近，蒙日安培偏微分方程的数值逼近，数值最优输运问题。该项目的具体目标包括(i)基于分段线性或节点函数构建新的数值方法，以保持离散极大值原理，这是这些问题的基本性质；（ii）将鲁棒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，以提高方法的准确性和效率；（iii）基于非线性PDE工具的离散版本对这些方法进行分析。(4)应用这些方法模拟液晶材料和天线设计。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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.