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An Interactive Introduction to the Finite Element Method

The Galerkin Method

1. Solving the Poisson Equation
2. Boundary Conditions
3. Variable Coefficients
4. Iterative Solvers
5. 3D Solid Mechanics
6. Exercises

Abstract Theory

7. Basic properties
8. Projection onto subspaces
9. Riesz representation theorem and symmetric variational problems
10. Coercive variational problems and their approximation
11. Inf-sup stable variational problems
12. Exercises

Sobolev Spaces

13. Generalized derivatives
14. Sobolev spaces
15. Trace theorems and their applications
16. Equivalent norms on \(H^1\) and on sub-spaces
17. The weak formulation of the Poisson equation
18. Experiments with norms
19. Exercises

Finite Element Method

20. Finite Element Method
21. Implementation of Finite Elements
22. Finite element system assembling
23. Implement our own system assembling
24. Finite element error analysis
25. Non-conforming Finite Element Methods

A posteriori error estimates

26. A posteriori error estimates
27. The residual error estimator
28. Goal driven error estimates
29. Equilibrated Residual Error Estimates

High Order Finite Elements

30. hp - Finite Elements
31. Implementation of High Order Finite Elements

Mixed Finite Element Methods

32. Stokes Equation
33. Boundary Conditions
34. Mixed Methods for second order equations
35. Abstract Theory
36. Abstract theory for mixed finite element methods
37. Parameter Dependent Problems

Discontinuous Galerkin Methods

38. Stationary Transport Equation
39. Instationary Transport Equation
40. Nitsche’s Method for boundary and interface conditions
41. DG - Methods for elliptic problems
42. Hybrid DG for elliptic equations
43. Splitting Methods for the time-dependent convection diffusion equation
44. Fourth Order Equation
45. H(div)-conforming Stokes

Mixed Methods for Second Order Equations

46. Application of the abstract theory
47. The function space \(H(\operatorname{div})\)
48. Finite Elements in \(H(\operatorname{div})\)
49. Finite Element Error Analysis
50. Error Analysis in \(L_2 \times H^1\)
51. Hybridization Techniques

Elasticity

52. Modeling Elasticity
53. Newton’s method
54. Solving nonlinear Elasticity
55. Linearized elasticity
56. Elastodynamics with Newmark time-stepping

Plates and Shells

57. Structural mechanics
58. Kirchhoff plate equation
59. Reissner Mindlin plates
60. Naghdi shell
61. Fully nonlinear Koiter shell model

Maxwell Equations

62. Maxwell equations
63. Magnetic flux induced by a coil
64. The function space H(curl)
65. The de`Rham complex

Time-dependent problems

66. Heat Equation
67. Wave Equation
68. Mass-lumping and Local time-stepping
69. Discontinuous Galerkin for the Wave Equation
70. Nano-optics: A ring-resonator

Iteration Methods

71. Basic Iterative Methods
72. The Richardson Iteration
73. The Gradient Method
74. Preconditioning
75. The Chebyshev Method
76. Conjugate Gradients

Sub-space Correction Methods

77. Additive Schwarz Methods
78. Some Examples of ASM preconditioners
79. high order fem
80. Domain Decomposition with minimal overlap
81. Overlapping Domain Decomposition Methods
82. Exercise: Robust preconditioners

Multigrid Methods

83. Multigrid and Multilevel Methods
84. Analysis of the multi-level preconditioner
85. Analysis of the Multigrid Iteration
86. Multi-level Extension
87. Algebraic Multigrid Methods
88. H(curl) - AMG

Saddle-point Problems

90. Structure of Saddle-point Problems
91. The Bramble-Pasciak Transformation
92. A Small Number of Constraints
93. Parameter Dependent Problems

Non-overlapping Domain Decomposition Methods

94. Introduction to Non-overlapping Domain Decomposition
95. Traces spaces
97. FETI methods
98. FETI-DP
99. BDDC - Preconditioner

Parallel Solvers

100. Introduction to MPI with mpi4py
101. Distributed Meshes and Spaces
102. Consistent and Distributed Vectors
103. Iteration methods in parallel
104. Using PETSc
105. NGSolve - PETSc interface
106. Solving Stokes in parallel

Index

By J. Schöberl

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