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

PI建议使用间断Galerkin(DG)方法对两个最困难的数值偏微分方程(PDE)问题进行全面的研究。该奖项的第一个主要目标是发展收敛的直接DG离散化方法来逼近一般二阶完全非线性偏微分方程组的粘性解，这是建立在PI先前对这些偏微分方程组发展间接数值方法的成功研究的基础上的。这部分研究项目的目标是：(I)将直接非标准DG方法推广到高维Monge-Ampere方程和Bellman方程；(Ii)建立DG方法的一般收敛理论；(Iii)开发有效的非牛顿非线性求解器来求解所得到的非线性系统；(Iv)将所得到的DG方法应用于完全非线性的PDE应用问题，包括最优质量传输问题、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DG有限元微积分理论。该奖项的第二个主要目标是开发绝对稳定、对求解器友好和保持矫顽力的DG离散化方法和高频声波、弹性和电磁波方程的两级Schwarz快速求解器。要分解高度振荡的波，必须使用足够精细的网格，这反过来又会导致要求解的庞大的代数系统。即使在今天的高性能计算机上，如果采用蛮力方法，也是因为计算量巨大，再加上高频波问题的强烈不确定性和极端病态的性质，使它们变得难以处理。克服挑战的最终解决方案必须在算法层面上寻求。这部分研究的目标是：(I)设计、分析和实现新的绝对稳定的、求解器友好的和保持矫顽力的DG离散化方法来求解这三类高频波动方程；(Ii)开发、分析和测试新的可并行的两层Schwarz解方法来求解由此产生的大型代数系统。该研究的完成将对新兴的数值完全非线性偏微分方程组和蓬勃发展的高频波计算领域产生重要的理论和实践影响。预期的新的数值能力可用于解决各种完全非线性的偏微分方程组问题和波散射问题，这些问题涉及微分几何、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、国防和电信以及金融行业。这项研究项目的教育部分是培养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。

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.