1. Introduction -- 2. The finite element method (FEM) -- 3. Mechanical field -- 4. Electromagnetic field -- 5. Acoustic field -- 6. Coupled electrostatic-mechanical systems -- 7. Coupled magnetomechanical systems -- 8. Coupled mechanical-acoustic systems -- 9. Piezoelectric systems -- 10. Multigrid solvers -- 11. Industrial applications -- 12. Summary and outlook -- A. Norms -- B. Scalar and vector fields -- C. Appropriate function spaces -- D. Solution of nonlinear equations