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

与有限元法一样，有限体积法也是一种求解偏微分方程的离散化方法。有限体积法由于具有局部守恒的特性以及对非结构网格的鲁棒性等优点，在计算流体力学中得到了广泛的应用。与有限元方法不同，有限体积法的数值分析是非常有限的。分析Navier-Stokes方程有限体积近似的论文屈指可数。本文的研究目标是建立一类求解Navier-Stokes方程的有限体积方法的完整数值分析方法，包括建立方法的稳定性；给出了最优先验误差估计、后验误差估计和解的超收敛性；证明了自适应过程的收敛性。计算流体力学涵盖了从飞机周围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代码中最常用的经典数值方法。该项目的目的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.