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

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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.