Numerical Methods and Algorithms for Fully Nonlinear Second Order Evolution Equations with Applications

完全非线性二阶偏微分方程（PDE）是指 一类非线性二阶偏微分方程，其在（至少一个）秒内是非线性的 未知函数的阶偏导数这样一类偏微分方程的产生， 科学和工程领域，包括天体物理学、微分几何学、地转 流体动力学，图像处理，动力学理论，材料科学，质量运输， 气象学和最优控制它们构成了最困难的一类偏微分方程， 分析和数值近似。在PI最近成功的基础上， 开发收敛和有效的数值方法和算法，完全非线性 二阶（时间无关）椭圆偏微分方程，拟议的研究项目打算 对数值方法和算法进行全面系统的研究 用于完全非线性二阶（时间相关）进化偏微分方程。的目标 建议的研究包括（i）发展消失矩方法和矩解 理论完全非线性二阶演化偏微分方程，（ii）发展完全离散 Galerkin型数值方法（例如有限元方法，混合有限元方法， 完全非线性二阶发展偏微分方程谱方法和间断Galerkin方法 基于消失矩方法，（iii）应用和/或调整开发的数值 方法，以一些新兴的应用问题，这是由完全非线性第二 为了偏微分方程，（四）开发计算机代码实现建议的数值方法。 作为完全非线性二阶发展偏微分方程的数值逼近， 子区域内的数值偏微分方程和那些偏微分方程出现了许多重要的应用， 天体物理学，微分几何，地转流体动力学，图像处理，动力学理论， 材料科学，大众运输，气象学，和最佳控制，完成了 拟议的研究项目预计将对解决这类偏微分方程产生深远的影响， 为解决一系列重要应用提供急需的能力和支持工具 这是由完全非线性二阶偏微分方程的问题。作为副产品， 溶液理论有望为我们理解粘度溶液提供一些启示 理论，并可能很有可能提供一个逻辑和自然的概括和扩展 从计算上看，粘性溶液理论既不自然，也不实用， 的观点该项目的教育部分是让研究生参与和培训， 发展必要的应用和计算数学知识和技能，使他们能够 在未来的科学和工程事业中取得成功。

Fully nonlinear second order partial differential equations (PDEs) are referred to a class of nonlinear second order PDEs which are nonlinear in (at least one) second order partial derivatives of unknown functions. Such a class of PDEs arise from many scientific and engineering fields including astrophysics, differential geometry, geostrophic fluid dynamics, image processing, kinetic theory, materials science, mass transportation, meteorology, and optimal control. They constitute the most difficult class of PDEs to analyze analytically and to approximate numerically. Building on the PI's recent success on developing convergent and efficient numerical methods and algorithms for fully nonlinear second order (time-independent) elliptic PDEs, the proposed research project intends to carry out a comprehensive and systematic study of numerical methods and algorithms for fully nonlinear second order (time-dependent) evolution PDEs. The objectives of the proposed research include (i) to develop the vanishing moment method and the moment solution theory for fully nonlinear second order evolution PDEs, (ii) to develop fully discrete Galerkin type numerical methods (e.g. finite element methods, mixed finite element methods, spectral and discontinuous Galerkin methods) for fully nonlinear second order evolution PDEs based on the vanishing moment methodology, (iii) to apply and/or to adapt the developed numerical methods to a number of emerging application problems which are governed by fully nonlinear second order PDEs, (iv) to develop computer codes for implementing the proposed numerical methods. As numerical approximations of fully nonlinear second order evolution PDEs is an untouched sub-area within the numerical PDEs and those PDEs arise from many important applications in astrophysics, differential geometry, geostrophic fluid dynamics, image processing, kinetic theory, materials science, mass transportation, meteorology, and optimal control, the completion of the proposed research project is expected to have a profound impact on solving this class of PDEs and on providing the much needed capability and enabling tools for solving a range of important application problems which are governed by fully nonlinear second order PDEs. As a by-product, the moment solution theory is expected to give some insights to our understanding of the viscosity solution theory, and might be very likely to provide a logical and natural generalization and extension for the viscosity solution theory which is not natural and neither practical from the computational point of view. The educational component of this project is to engage and train graduate students in developing necessary applied and computational mathematics knowledge and skills so that they can pursue a successful career in science and engineering in the future.