Discontinuous Galerkin and Hybridized Methods for Partial Differential Equations

这个项目致力于设计和研究，从理论上和计算上，有效的方法来数值解决偏微分方程建模的问题。我们集中精力研究所谓的不连续伽辽金方法和杂化混合方法在各种实际问题中的应用所带来的不同困难。这些应用包括曲面的演化和机器人在任意地形中的导航（Hamilton-Jacobi方程）、不可压缩流体的流动（Navier-Stokes方程）、特别强调壳的结构力学（弹性方程）、电磁波传播（Maxwell方程）和多孔介质中的达西流动（二阶椭圆方程）。对于上述所有问题，我们考虑的问题是如何获得高精度的计算机模拟。在许多这类问题中，重点将放在适当地设计不连续伽辽金方法和杂交方法，使它们比以前已知的方法更加鲁棒和高效。在其他情况下，努力将集中在模拟质量的有效控制上。事实上，由于这些复杂问题的确切解是未知的，为了保证模拟的给定准确性，必须设计特殊技术以评估其质量。此外，这些技术可用于自动让计算机知道何时何地增加或减少计算工作量以获得仿真；这样大大提高了其计算效率。不连续伽辽金方法特别适合这种方法。

This project is devoted to devising and studying, theoretically as well as computationally, efficient methods for numerically solving problems modeled by partial differential equations. We focus our efforts on the different difficulties raised by the application of the so-called discontinuous Galerkin method and the hybridized mixed methods to a variety of problems of practical interest. The applications are to the evolution of surfaces and to navigation of robots in arbitrary terrains (Hamilton-Jacobi equations), incompressible fluid flow (Navier-Stokes equations), structural mechanics with special emphasis on shells (elasticity equations), to electromagnetic wave propagation (Maxwell equations), and to Darcy flow in porous media (second-order elliptic equations).For all of the above problems, we consider the problem of how to obtain highly accurate computer simulations. In many of these problems, the emphasis will be put on the suitable devising of the discontinuous Galerkin and hybridized methods so that they become more robust and efficient than previously known methods. In others, the effort will be concentrated in the efficient control of the quality of the simulation. Indeed, since the exact solution of these complex problems is not known, to guarantee a given accuracy of the simulation, special techniques have to be devised in order to assess its quality. Moreover, these techniques can be employed to automatically let the computer know when and where to increase or decrease the computational effort to obtain the simulation; in this way, the efficiency of its computation is significantly enhanced. Discontinuous Galerkin methods are particularly well suited for this approach.