Analisys of Numerical Differential Equations and Finite Element Method (J. BrandenBurg).pdf

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First Edition, 2012
ISBN 978-81-323-1362-5
© All rights reserved.
Published by:
College Publishing House
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Ansari Road, Darya Ganj,
Delhi - 110002
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Table of Contents
Chapter 1 - Numerical Ordinary Differential Equations
Chapter 2 - Boundary Element Method, Beeman's Algorithm and Adaptive
Stepsize
Chapter 3 - Céa's Lemma
Chapter 4 - Constraint Algorithm
Chapter 5 - Compact Stencil, Courant–Friedrichs–Lewy Condition and
Direct multiple Shooting Method
Chapter 6 - Crank–Nicolson Method
Chapter 7 - Discrete Laplace Operator and Discrete Poisson Equation
Chapter 8 - Euler Method
Chapter 9 - Finite Difference
Chapter 10 - Finite Difference Method
Chapter 11 - Finite Element Method
Chapter 12 - Bramble-Hilbert Lemma and Spectral Element Method
Chapter 13 - hp-FEM
Chapter 14 - Finite Element Method in Structural Mechanics
Chapter 15 - Interval Finite Element
Chapter 16 - Modal Analysis using FEM
Chapter 17 - Domain Decomposition Methods and Additive Schwarz
Method
Chapter 1
Numerical Ordinary Differential
Equations
Illustration of numerical integration for the differential equation y ' = y , y (0) = 1. Blue: the
Euler method, green: the midpoint method, red: the exact solution, y = e t . The step size is
h = 1.0.
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The same illustration for h = 0.25. It is seen that the midpoint method converges faster
than the Euler method.
Numerical ordinary differential equations is the part of numerical analysis which
studies the numerical solution of ordinary differential equations (ODEs). This field is also
known under the name numerical integration, but some people reserve this term for the
computation of integrals.
Many differential equations cannot be solved analytically, in which case we have to
satisfy ourselves with an approximation to the solution. The algorithms studied here can
be used to compute such an approximation. An alternative method is to use techniques
from calculus to obtain a series expansion of the solution.
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