Chapter 10
Hyperbolic Problems
10.1 Conservation Laws
We have successfully applied nite element methods to elliptic and parabolic problems
however, hyperbolic problems will prove to be more di cult. We got an inkling of this
while studying convection-di u
Chapter 9
Parabolic Problems
9.1 Introduction
The nite element method may be used to solve time-dependent problems as well as
steady ones. This e ort involves both parabolic and hyperbolic partial di erential systems. Problems of parabolic type involve di
Chapter 8
Adaptive Finite Element Techniques
8.1 Introduction
The usual nite element analysis would proceed from the selection of a mesh and basis
to the generation of a solution to an accuracy appraisal and analysis. Experience is the
traditional method
Chapter 7
Analysis of the Finite Element
Method
7.1 Introduction
Finite element theory is embedded in a very elegant framework that enables accurate a
priori and a posteriori estimates of discretization errors and convergence rates. Unfortunately, a large
Chapter 6
Numerical Integration
6.1 Introduction
After transformation to a canonical element 0 , typical integrals in the element sti ness
or mass matrices (cf. (5.5.8) have the forms
ZZ
Q=
( )NsNT det(Je)d d
(6.1.1a)
t
0
where ( ) depends on the coe cien
Chapter 5
Mesh Generation and Assembly
5.1 Introduction
There are several reasons for the popularity of nite element methods. Large code segments can be implemented for a wide class of problems. The software can handle complex
geometry. Little or no softw
Chapter 4
Finite Element Approximation
4.1 Introduction
Our goal in this chapter is the development of piecewise-polynomial approximations U
of a two- or three-dimensional function u. For this purpose, it su ces to regard u as
being known and to determine
Chapter 3
Multi-Dimensional Variational
Principles
3.1 Galerkin's Method and Extremal Principles
The construction of Galerkin formulations presented in Chapters 1 and 2 for one-dimensional
problems readily extends to higher dimensions. Following our prior
Chapter 2
One-Dimensional Finite Element
Methods
2.1 Introduction
The piecewise-linear Galerkin nite element method of Chapter 1 can be extended in
several directions. The most important of these is multi-dimensional problems however,
we'll postpone this
Chapter 1
Introduction
1.1 Historical Perspective
The nite element method is a computational technique for obtaining approximate solutions to the partial di erential equations that arise in scienti c and engineering applications. Rather than approximating