Title

The Finite Element Approximation of Convection-Diffussion-Type Equations

Year

2010

Season

Spring

Paper Type

Master's Thesis

College

College of Arts and Sciences

Degree Name

Master of Science in Mathematical Sciences (MS)

Department

Mathematics & Statistics

Abstract

In this thesis we study convection-diffusion equations using numerical methods. The convection-diffusion equation is an elliptic partial differential equation, which describes physical phenomena where particles or energy (or other physical quantities such as heat) are transferred inside a physical system due to two processes: diffusion and convection. Convection is the transfer of heat by the circulation or movement of parts of a liquid or gas. While diffusion is the process in which there is movement of a substance from an area of high concentration of that substance to an area of lower concentration. Convection-diffusion equations are employed in various branches of physics and engineering. For example, the transport of air and ground water pollutants and oil reservoir flow can be modeled by using the convection-diffusion equation. Also, a wide variety of applications are found in fluid mechanics. We discuss both variational formulation and the finite element approximation using linear triangular elements. Then we apply these methods to solve convection-diffusion equations with Dirichlet boundary conditions. Although the primary focus of this study is to obtain an accurate approximation to the solution to the convection-diffusion equation, we discuss an application to the non–linear Burgers Equation. The non linear Burgers equation is linearized and the solution is computed using an iterative procedure. Each iteration is accomplished by solving two convection–diffusion equations are solved.

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