Novel Discontinuous Galerkin Finite Element Methods for Second Order Fully Nonlinear Equations and High Frequency Wave Equations

二阶完全非线性方程和高频波动方程的新型间断伽辽金有限元方法

• 批准号: 1318486
• 负责人: Xiaobing Feng
• 金额: $ 26万
• 依托单位: University of Tennessee Knoxville
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1318486&HistoricalAwards=false
• 关键词: Novel; Discontinuous; Galerkin; Finite; Element

The PI proposes to carry out a comprehensive study for two of most difficult numerical partial differential equation (PDE) problems using discontinuous Galerkin (DG) methods. The first main goal of the award is to develop convergent direct DG discretization methods for approximating viscosity solutions of general second order fully nonlinear PDEs, which builds upon the PI's previous successful research on developing indirect numerical methods for these PDEs. The objectives of this part of the research project are: (i) to extend the direct nonstandard DG methods to high-dimensional Monge-Ampere and Bellman equations; (ii) to establish a general convergent theory for the proposed DG methods; (iii) to develop efficient non-Newtonian nonlinear solvers for solving the resulting nonlinear systems; (iv) to apply the resulting DG methods to fully nonlinear PDE application problems including the optimal mass transport problem, the semigeostrophic flow problem, and stochastic optimal control problems; (v) to further develop the DG finite element differential calculus theory resulted from the proposed research project. The second main goal of the award is to develop absolutely stable, solver-friendly, and coercivity-preserving DG discretization methods and two-level Schwarz fast solvers for high frequency acoustic, elastic and electromagnetic wave equations. To resolve highly oscillatory waves, sufficiently fine mesh must be used, which in turn results in huge algebraic systems to solve. It is the sheer amount of computations coupled with the strong indefiniteness and the extremely ill-conditioned nature of high frequency wave problems that makes them intractable even on today's high performance computers if the brute force approach is adopted. The ultimate solution to overcome the challenge must be sought at the algorithmic level. The objectives of this part of the research project are: (i) to design, analyze and implement novel absolutely stable, solver-friendly, and coercivity-preserving DG discretization methods for the three types of high frequency wave equations; (ii) to develop, analyze and test novel parallelizable two-level Schwarz solution methods for solving the resulting large algebraic systems.The completion of the proposed research will have a significant theoretical and practical impact on the emerging field of numerical fully nonlinear PDEs and the thriving field of high frequency wave computation. The anticipated new enabling numerical capabilities can be used to solve various fully nonlinear PDE problems and wave scattering problems arising from differential geometry, antenna design, astrophysics, geophysical fluid dynamics, image processing, optimal control and optimal mass transport, petroleum engineering, geoscience, medical science, defense and telecommunication as well as financial industries. The education component of this research project is train graduate students in developing necessary applied and computational mathematics knowledge and skills so that they can pursue a successful career in either academia or industry in the near future.

PI建议使用不连续伽辽金（DG）方法对两个最困难的数值偏微分方程（PDE）问题进行综合研究。该奖项的第一个主要目标是开发收敛直接 DG 离散化方法，用于逼近一般二阶完全非线性偏微分方程的粘度解，该方法建立在 PI 之前为这些偏微分方程开发间接数值方法的成功研究的基础上。这部分研究项目的目标是： (i) 将直接非标准 DG 方法扩展到高维 Monge-Ampere 和 Bellman 方程； (ii) 为所提出的 DG 方法建立通用收敛理论； (iii) 开发高效的非牛顿非线性求解器来求解所得的非线性系统； (iv) 将所得的 DG 方法应用于完全非线性 PDE 应用问题，包括最优质量传递问题、半地转流问题和随机最优控制问题； (v) 进一步发展所提出的研究项目所产生的 DG 有限元微分理论。该奖项的第二个主要目标是开发绝对稳定、求解器友好且保持矫顽力的 DG 离散方法和用于高频声波、弹性波和电磁波方程的两级 Schwarz 快速求解器。为了解决高度振荡的波，必须使用足够细的网格，这反过来又导致需要解决巨大的代数系统。高频波问题的计算量巨大，加上强烈的不确定性和极其病态的性质，使得即使在当今的高性能计算机上，如果采用强力方法，它们也很难解决。必须在算法层面寻求克服挑战的最终解决方案。这部分研究项目的目标是：（i）针对三种类型的高频波方程设计、分析和实现绝对稳定、求解器友好且保持矫顽力的新型 DG 离散化方法； (ii) 开发、分析和测试新颖的可并行两级 Schwarz 求解方法，用于求解由此产生的大型代数系统。所提出的研究的完成将对数值完全非线性偏微分方程和偏微分方程的新兴领域产生重大的理论和实践影响高频波计算领域蓬勃发展。预期的新的启用数值功能可用于解决微分几何、天线设计、天体物理学、地球物理流体动力学、图像处理、最优控制和最优质量传输、石油工程、地球科学、医学等领域产生的各种完全非线性偏微分方程问题和波散射问题科学、国防和电信以及金融行业。该研究项目的教育部分是培训研究生发展必要的应用和计算数学知识和技能，以便他们能够在不久的将来在学术界或工业界追求成功的职业生涯。