Superconvergent Discontinuous Galerkin methods for Partial Differential Equations

在这个提案中，研究者研究和开发了新一代的不连续Galerkin方法，其特点是更容易实现，具有更好的稳定性和收敛特性，并显示出处理任意形状区域的更好的灵活性。调查员将把精力集中在四个特定的问题上。第一个是在流体流动领域，包括建立一个非常有竞争力的不可压缩Navier-Stokes方程的数值方法的一般理论。二是在连续介质力学领域，研究四阶问题的最优收敛方法，为非线性壳体数值方法的设计铺平道路。第三是在非线性守恒律领域，包括引入新的技术，以克服十多年来拖累这些有用方程的高效、高精度方法发展的两个主要困难。最后是在处理曲线边界的技术领域，包括用包含区域的非常简单的长方体网格并在其边界附近使用特殊的近似技术来取代传统的高精度网格化区域的范例。物理现象的计算机模拟在工程和物理中的广泛应用中是一个具有高度实用价值的工具。研究人员研究了一种新兴的、有前途的技术，用高精度和更有效的算法来执行这些模拟，以解决广泛的实际感兴趣的问题。它们包括在航空航天和力学(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)中的许多应用。

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).