A FEniCS Tutorial - FEniCS Project

Contents
1 Fundamentals 4
1.1 The Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Controlling the Solution Process . . . . . . . . . . . . . . . . . . 15
1.5 Linear Variational Problem and Solver Objects . . . . . . . . . . 17
1.6 Examining the Discrete Solution . . . . . . . . . . . . . . . . . . 17
1.7 Solving a Real Physical Problem . . . . . . . . . . . . . . . . . . 20
1.8 Quick Visualization with VTK . . . . . . . . . . . . . . . . . . . 24
1.9 Computing Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 26
1.10 A Variable-Coefficient Poisson Problem . . . . . . . . . . . . . . 28
1.11 Computing Functionals . . . . . . . . . . . . . . . . . . . . . . . 31
1.12 Visualization of Structured Mesh Data . . . . . . . . . . . . . . . 35
1.13 Combining Dirichlet and Neumann Conditions . . . . . . . . . . 38
1.14 Multiple Dirichlet Conditions . . . . . . . . . . . . . . . . . . . . 42
1.15 A Linear Algebra Formulation . . . . . . . . . . . . . . . . . . . . 43
1.16 Parameterizing the Number of Space Dimensions . . . . . . . . . 46
2 Nonlinear Problems 48
2.1 Picard Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 A Newton Method at the Algebraic Level . . . . . . . . . . . . . 51
2.3 A Newton Method at the PDE Level . . . . . . . . . . . . . . . . 54
2.4 Solving the Nonlinear Variational Problem Directly . . . . . . . . 55
3 Time-Dependent Problems 58
3.1 A Diffusion Problem and Its Discretization . . . . . . . . . . . . 59
3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Avoiding Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 A Physical Example . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Creating More Complex Domains 71
4.1 Built-In Mesh Generation Tools . . . . . . . . . . . . . . . . . . . 71
4.2 Transforming Mesh Coordinates . . . . . . . . . . . . . . . . . . . 72
5 Handling Domains with Different Materials 73
5.1 Working with Two Subdomains . . . . . . . . . . . . . . . . . . . 74
5.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Multiple Neumann, Robin, and Dirichlet Condition . . . . . . . . 77
6 More Examples 81
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