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

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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.