Superconvergent Discontinuous Galerkin methods for Partial Differential Equations

在本提案中，研究者研究并开发了新一代的不连续伽辽金方法，其特点是更容易实现，具有增强的稳定性和收敛性，并且在处理任意形状的域时表现出更好的灵活性。研究者将把他的精力集中在四个特定的问题上。第一个是在流体流动领域，包括为不可压缩的Navier-Stokes方程建立一个非常有竞争力的数值方法的一般理论。第二类是连续介质力学领域，研究四阶问题的最优收敛方法，为设计非线性壳的数值方法铺平道路。第三个是在非线性守恒定律领域，包括引入新技术，以克服十多年来阻碍这些有用方程的高效、高阶精确方法发展的两个主要困难。最后一个是处理曲面边界的技术领域，它包括用一个非常简单的包含域的盒子网格和在其边界附近使用特殊逼近技术的新方法取代传统的高精度域网格划分范式。物理现象的计算机模拟在工程和物理学的各种应用中是一种非常有价值的实用工具。研究者研究了一种新兴的、有前途的技术，用高精度和更有效的算法来进行这些模拟，以解决广泛的实际问题。它们包括许多应用于航空航天和力学（不可压缩流体流动，亚音速和超音速流动）以及土木工程（固体结构）。

In this proposal, the investigator studies and develops a new, emerging generation of discontinuous Galerkin methods characterized by being easier to implement, by having enhanced stability and convergence properties, and by displaying an improved flexibility for handling arbitrarily-shaped domains. The investigator will focuses his effort in four particular problems. The first is in the area of fluid flow and consists in establishing a general theory of very competitive numerical methods for the incompressible Navier-Stokes equations. The second is in the area of continuum mechanics and consists in the study of optimally convergent methods for fourth-order problems in order to pave the way to the devising of numerical methods for non-linear shells. The third is in the area of non-linear conservation laws and consists in the introduction of new techniques geared towards overcoming the two main difficulties that have dragged down for more than a decade the development of efficient, high-order accurate methods for these useful equations. The last is in the area of techniques for handling curved boundaries and consists in replacing the traditional paradigm of meshing the domain with high accuracy by the new approach of using a very simple mesh of a box containing the domain and employing special approximation techniques near its border.The computer simulation of physical phenomena is a highly valued tool of practical interest in a wide variety of applications in Engineering and Physics. The investigator studies an emerging and promising technique of carrying out these simulations with highly accurate and more efficient algorithms for a wide range of problems of practical interest. They include many applications to Aerospace and Mechanics (incompressible fluid flow, subsonic and supersonic flow) as well as to Civil Engineering (solid structures).