Superconvergent Hybridizable Discontinuous Galerkin and Mixed Methods for Partial Differential Equations

物理现象的计算机模拟在科学和工程的各种应用中具有很高的实用价值。该项目研究了两种新的、有前途的技术，以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）。本研究项目旨在引入一种系统的方法来获得新的、竞争的不连续伽辽金和混合方法，这些方法在由任意形状的元素组成的非结构化网格上超收敛。这将用于流体力学（包括不可压缩的Navier-Stokes方程）和连续介质力学（包括大变形弹性方程）中出现的各种线性和非线性偏微分方程。该项目还将考虑伴随恢复方法，这将导致一种非常有效的方法，比以前认为的一般有限元近似可能获得更高的精度。只需加倍计算量，逼近的精度阶数就会加倍。特别地，将这种技术应用于满足伽辽金正交性的方法将使精度的阶数提高四倍。

Computer simulation of physical phenomena is a highly valued tool of practical importance in a wide variety of applications in science and engineering. This project studies two new, promising techniques 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). This research project aims to introduce a systematic way of obtaining new, competitive discontinuous Galerkin and mixed methods that superconverge on unstructured meshes made of elements of arbitrary shape. This will be done for a wide variety of partial differential equations arising in fluid dynamics (including the incompressible Navier-Stokes equations) and continuum mechanics (including the equations of large deformation elasticity), both linear and nonlinear. The project will also consider adjoint-recovery methods that will result in a very efficient way of obtaining more accuracy than previously thought possible from general finite element approximations. By only doubling the computational effort, the order of accuracy of the approximation will be doubled. In particular, the application of this technique to methods satisfying a Galerkin orthogonality property will result in the quadrupling of the order of accuracy.