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

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的直接DG离散方法，用于近似一般二阶全非线性偏微分方程的粘度解，该方法建立在PI先前成功的研究基础上，为这些偏微分方程开发了间接数值方法。本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）为建议的DG方法建立一般收敛理论；（iii）开发有效的非牛顿非线性解算器来求解由此产生的非线性系统；（iv）将所得的DG方法应用于完全非线性的PDE应用问题，包括最优质量传递问题、半转流问题和随机最优控制问题；(v)进一步发展由建议研究计划所产生的DG有限元微分理论。该奖项的第二个主要目标是为高频声波、弹性和电磁波方程开发绝对稳定、求解器友好且保压的DG离散化方法和两级Schwarz快速求解器。为了解决高振荡波，必须使用足够精细的网格，这反过来又导致了巨大的代数系统来求解。如果采用蛮力方法，那么即使在今天的高性能计算机上，高频波问题的计算量，加上强不确定性和极端病态的性质，也会使它们变得难以处理。克服这一挑战的最终解决方案必须在算法层面上寻求。本课题的研究目标是：(1)设计、分析和实现三种高频波动方程的绝对稳定、求解器友好、保压的DG离散化新方法；（ii）开发，分析和测试新的可并行的两级Schwarz解决方法，以解决由此产生的大型代数系统。本研究的完成将对新兴的数值全非线性偏微分方程领域和蓬勃发展的高频波计算领域产生重大的理论和实际影响。预期的新数字能力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运输、石油工程、地球科学、医学、国防和电信以及金融行业中出现的各种全非线性PDE问题和波散射问题。这个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.