# The Finite Element Method

#### Lecturers

Dr Bishnu Lamichhane, The University of Newcastle

#### Synopsis

The finite element method is one of the most powerful techniques in approximating the solution of partial differential equations arising in the mathematical modelling of many physical and engineering processes. The finite element method is based on firm mathematical foundation, where mathematical tools from functional analysis, approximation theory and variational calculus are applied to analyse the whole approximation process. This course aims at introducing the theory and computation of finite element techniques for elliptic and parabolic partial differential equations. Applications to heat transfer, elasticity and image processing will be discussed.

#### Course Overview

This course will provide an introduction to the finite element method. The following topics will be covered:

• Weak formulation: Weak formulation of partial differential equations, Sobolev spaces, well-posedness, finite element discretisation, error estimates
• Finite element spaces: Linear and quadratic finite element methods in one, two and three dimensions, construction of some finite element spaces, Reaction-diffusion and convection-reaction-diffusion problems
• Implementation: Programming linear and quadratic finite element methods for convectionreaction-diffusion problems, time-dependent problems
• Applications: heat transfer, image processing and solid mechanics

#### Contact Hours

28 hours of lectures spread over the four weeks, with consultation as requested/required. Some lectures may be held in a computer lab.

#### Prerequisites

• Analysis and differential equations: Mathematical analysis and differential equations of second year level.
• MATLAB: Background in programming in MATLAB.

#### Assessment

• Two assignments 50%
• Final exam 50%

Most of the materials will be based on following finite element books:

• D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Third Edition, Cambridge University Press, 2007
• B.D. Reddy, Introductory Fucntional Analysis with Applications to Boundary Value Problems and Finite Elements, Springer, 1998
• P.G. Ciarlet, The Finite Element Method for Elliptic Problems, SIAM, Philadelphia, 2002

Lecture notes will be provided on a weekly basis.