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）问题进行全面研究。该奖项的第一个主要目标是开发收敛的直接DG离散化方法，用于近似一般二阶完全非线性偏微分方程的粘性解，这是建立在PI以前成功研究开发这些偏微分方程的间接数值方法的基础上。这部分研究的目标是：（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）发展有效的非牛顿非线性求解器，用于求解所得到的非线性方程组，（iv）建立一个新的非线性方程组求解器，（iv）建立一个新的非线性方程组求解器。（iv）将所得到的DG方法应用于完全非线性PDE应用问题，包括最优质量输送问题、半地转流问题和随机最优控制问题;（v）进一步发展本研究项目所产生的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的，求解器友好的，并保持连续性的DG离散化方法和两级施瓦茨快速求解器的高频声波，弹性和电磁波方程。为了解决高振荡波，必须使用足够细的网格，这反过来又导致巨大的代数系统来解决。这是纯粹的计算量加上强大的不确定性和极病态的性质，使他们棘手的高频波问题，即使在今天的高性能计算机，如果蛮力的方法是通过。克服这一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、分析和实现绝对稳定的、求解器友好的、保持离散性的三类高频波动方程离散方法; ㈡发展，分析和测试新的并行两个-水平施瓦茨求解方法求解由此产生的大型代数系统。完成拟议的研究将有重要的理论意义和实际影响的新兴领域的数值完全非线性偏微分方程和蓬勃发展的领域的高频波计算。预期的新的使数值能力可用于解决各种完全非线性偏微分方程问题和波散射问题所产生的微分几何，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。本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，使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.