Date of Award

1-1-2012

Document Type

Campus Access Dissertation

Department

Mathematics

First Advisor

Ju, Lili

Abstract

In this dissertation we study covolume-upwind finite volume methods on rectilinear grids for solving second-order linear elliptic partial differential equations with mixed boundary conditions. This class of problems has intensive applications in various fields of science and engineering, e.g., fluid mechanics, groundwater prediction, environmental protection. An advantage of the finite volume method is that one can convert integrations over local control volumes (also called covolumes) to integrations over their boundaries based on the Green's theorem. Finite difference or quadrature rules then can be used to approximate the resulting terms in local elements. The discretization schemes derived from finite volume methods usually satisfy local mass conservation, which makes the finite volume method an important and effective tool to numerically simulate various physical models. To avoid non-physical numerical oscillations for convection-dominated problems, nonstandard control volumes (covolumes) are generated based on local Peclet's numbers and the upwind principle for the finite volume approximations. Two types of discretization schemes with mass lumping are developed, using bilinear or biquadratic basis functions as the trial space respectively. Some stability analysis of the schemes are presented for the model problem with constant coefficients. Various examples in both two dimensions and three dimensions are also carried out to numerically demonstrate stability and optimal convergence of the proposed methods.

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