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Contents

Series Preface v

Preface to the English Edition vii

Preface to the German Edition xi

0 For Example: Modelling Processes in Porous Media
with Differential Equations 1
0.1 The Basic Partial Differential Equation Models . . . . . 1
0.2 Reactions and Transport in Porous Media . . . . . . . . 5
0.3 Fluid Flow in Porous Media .. .. ... .. .. ... .. 7
0.4 Reactive Solute Transport in Porous Media . . . . . . . . 11
0.5 Boundary and Initial Value Problems . . . . . . . . . . . 14

1 For the Beginning: The Finite Difference Method
for the Poisson Equation 19
1.1 The Dirichlet Problem for the Poisson Equation . . . . . 19
1.2 The Finite Difference Method . .. ... .. .. ... .. 21
1.3 Generalizations and Limitations
of the Finite Difference Method .. ... .. .. ... .. 29
1.4 Maximum Principles and Stability . . . . . . . . . . . . . 36

2 The Finite Element Method for the Poisson Equation 46
2.1 Variational Formulation for the Model Problem . . . . . 46