1. Lecture 9: Finite Elements Sauro Succi

2. FEM: non-spherical cows Coordinate-free: Unstructured

3. FEM for fluids

4. The Finite Element Method

5. Less intuition, more systematic, solid math Foundations (functional analysis) Strong vs Weak Convergence

6. Pointwise (strong) formulation Local interpolation around x=x_j: Looses accuracy on non-uniform meshes Awkward on unstructured lattices

7. Compute gradient below?

8. Variational (weak) formulation Hilbert space L2: Global statement For any g, find f_N such that:

9. Expansion on basis function Convergence in Hilbert space Projection on Hilbert space

10. Expansion on basis function Operators to Matrices

11. Examples of matrices: Mass, Stiffness, Advection. using linear hat functions

12. FEM matrix operators

13. Finite-support basis function

14. Useful identities

15. Mass matrix Uniform mesh:

16. Mass matrix: smoother

17. Advection matrix Uniform mesh:

19. Diffusion matrix Uniform mesh:

20. Self-advection matrix: triad Uniform mesh:

21. FEM operators

22. Matrix assembly Element-wise

23. FEM operators + Strong math back-up - Expensive (matrix algebra) + Very systematic + Fluid/Solid coupling

24. Summary FEM +: Powerful math backup (weak convergence) Systematic programming Geometrical flexibility FEM -: Matrix algebra anyway (lumping) Heavy duty Mainstream for solid mech, not fluids

25. 1d example: assembly 4 matrix elements per interval; 2 intervals per node= 8 matrix elements/node

26. FEM: cows are cows Coordinate-free: Unstructured

29. Boundary conditions Element-wise

30. Triangle basis function

32. Matrix assembly Element-wise: connectivity

33.        Matrix assembly

34. Linear Algebra Direct Methods: Minimize bandwidth Optimal Numbering (NP complete) Iterative Methods: Sparse matrix algebra: A*x+y

35. Optimal Numbering

37. Some app’s from the web + sample code fem.f

38. Finite-support basis function