A complete numerical analysis for finite volume methods to The Navier-Stokes equations

与有限元法一样，有限体积法是一种求解偏微分方程的离散化技术。由于有限体积法具有局部守恒性和非结构网格鲁棒性等其他吸引人的性质，因此在计算流体力学中得到了广泛的应用。与有限元方法不同，有限体积方法的数值分析非常有限。本文的主要目的是对Navier-Stokes方程的一类有限体积方法建立一个完整的数值分析，包括建立方法的稳定性，得到最优的先验误差估计，后验误差估计和解的超收敛性，并给出一个数值结果。并证明自适应过程的收敛性。计算流体力学涵盖了从飞机周围的空气流动到人体血液流动的广泛问题。任何计算流体动力学问题的数学基础都是Navier-Stokes方程。 有限体积法是商业软件和研究程序中最常用的经典数值方法。该项目的目标是为有限体积法提供数学理论。

Like the finite element method, the finite volume method is a discretization technique for solving partial differential equations. Due to the local conservation property and other attractive properties such as robustness with unstructured meshes, the finite volume method is widely used in computational fluid dynamics. Unlike finite element methods, numerical analysis for the finite volume methods is very limited. Only handful of papers analyzing the finite volume approximation for the Navier-Stokes equations can be found. The goal of the proposed research is to establish a complete numerical analysis for a class of the finite volume methods to the Navier-Stokes equations, including establishing stability of the methods; deriving optimal priori error estimates, posteriori error estimates and superconvergence of the solutions; and proving convergence of adaptive procedures. Computational fluid dynamics covers wide range of problems from air flow around airplane to blood flow of human body. The mathematical foundations of any computational fluid dynamics problem are the Navier-Stokes equations. The finite volume method is the classical numerical method for the problem used most often in commercial software and research codes. The objective of the project is to provide mathematical theory for the finite volume method.