The following books may be useful for learning more about the finite element method. • An Introduction to the Finite Element Method by Reddy [Red93]
12.4 Further Reading 159
Figure 12.2: FEM solution to the model problem (Laplaces equation) using a second-order Lagrange function space. The solution was visualized using Visit.
• Finite Element Methods for Flow Problems by Donea and Huerta [DH03]
• The Finite Element Method in Heat Transfer and Fluid Dynamics by Reddy and Gartling [RG94] • The Mathematical Theory of Finite Element Methods by Brenner and Scott [BS02]
Bibliography
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Index
A Antoine’s equation . . . 76 array construction . . . 28 array operations . . . 30 array slicing . . . 29 axis labels . . . 32 B backward difference approximation . . . 110backward substitution . . . 56 Bisection Method . . . 83 boundary conditions . . . 131, 132, 137 Broyden’s Method . . . 87 broyden1 function . . . 82, 93 C centered difference approximation . . . 110
Clausius-Clayperon equation . . . 70 colorbar . . . 33 comparators . . . 23 computational cost . . . 58 computational scalability . . . 58 conditionals . . . 23 containers . . . 20 csv files . . . 97 curve fitting . . . 75 D def keyword . . . 26 direct methods . . . 54 distillation column . . . 52 E Equation classification . . . 10 Error Messages . . . 34
explicit time stepping . . . 121
F figure title . . . 32
file, data import from . . . 97
finite difference method . . . 136
finite element mesh . . . 156
first derivative approximation . . . 110
for loops . . . 25
fsolve function . . . 93 function space . . . 156 G Gauss-Seidel iteration . . . 62 Gaussian elimination . . . 55 Gaussian quadrature . . . 116 Getting Python . . . 17 H histogram . . . 100 I if statements . . . 23 input function . . . 25 iterative methods . . . 59 J Jacobi iteration . . . 60 K kinetics, enzyme . . . 119 kinetics, first-order . . . 104 kinetics, Michaelis-Menton . . . 120 kinetics, second-order . . . 104 L Laplace’s equation . . . 141 linear regression . . . 71
linear regression statistics . . . 103
lists . . . 20 M Maple . . . 40 math library . . . 22 Mathcad . . . 40 Mathematica . . . 40 MATLAB . . . 40 Matplotlib . . . 18, 31 mean vector . . . 101 meshgrid . . . 33 midpoint rule . . . 114
modified Euler method . . . 123
N NaN . . . 83
newton krylov function . . . 93
Newton’s Method . . . 86
normal equations . . . 72
numerical integration . . . 113
numpy . . . 17, 28, 53 numpy linear algebra library . . . 54
P predator-prey models . . . 126 pylab . . . 31 pylab.plot() . . . 31 pyplot . . . 31 Python functions . . . 26 Python installation . . . 18 Python versions . . . 17 R range() function . . . 25 S Sage . . . 40
scipy . . . 18
scipy initial value problem . . . 125
scipy integrate library . . . 125
Scipy quadrature function . . . 116
Scipy Statistics . . . 102
secant method . . . 88
second derivative approximation . . . 112
second-order initial value problems . . . 126
Shooting Method . . . 132
Spyder . . . 18
SRK equation of state . . . 81
standard deviation . . . 101
standard score . . . 102
stiff differential equations . . . 126
symbolic derivatives . . . 46
symbolic integration . . . 47
symbolic mathematics . . . 39
SymPy . . . 18, 40 SymPy factor function . . . 44
SymPy solve function . . . 41
system of linear equations . . . 51
systems of nonlinear equations . . . 89
T t-test . . . 102
transient PDEs . . . 149
trapezoid rule . . . 114
V van der Waals equation . . . 42
vapor pressure . . . 70 vector norms . . . 60 vector summation . . . 73 W weak form . . . 157 Why Python? . . . 15 Wolfram Alpha . . . 40 Z z-score . . . 102