Finite element error analysis: an elementary approach

Abstract

The present dissertation entitled, “Finite Element Error Analysis: An Elementary Approach”, embodies a brief account of investigations carried out by various authors on finite element error analysis under the supervision of Dr. Vivek, Assistant Professor, School of Mathematics and Computer Applications, Thapar University, Patiala. The work presented in this dissertation has been divided into three chapters. The aim of this work is to study some results on error bounds in finite element methods. In the very first chapter there is a brief introduction about the finite element methods. Differential equations arise in the mathematical modelling of many physical, chemical and biological phenomena and many more areas of science and engineering. Asymptotic and numerical are two principle approaches for solving differential equations. In the present study, the numerical techniques have been used to approximate the solution. As there are many numerical methods to find an approximate solution such as finite difference methods(FDM), finite element methods(FEM), finite volume methods(FVM), boundary element methods(BEM), etc. The present work consists of finite element analysis. In the first chapter, finite element method has been discussed. The finite element method is one of the most powerful method known for finding the numerical solutions. In the first chapter, the method is outlined with the help of a simple example of finding the area of a circle. Then finite element errors are discussed. The errors can be categorized into two types. One is a priori error estimates and the other is a posteriori error estimates. Then the one dimensional model problem Lϕ = −aϕxx + bϕx + cϕ = f(x) in Ω =]0, 1[