1. BVP Weak Formulation Weak Formulation ( variational formulation) where Multiply equation (1) by and then integrate over the domain Green’s theorem gives

2. Green’s First identity in R^2 (p285) Green’s First Identity

3. Galerkin Methods Weak Formulation ( variational formulation) Infinite dimensional space Is finite dim Unique sol?

4. Galerkin Methods Discrete Form Is finite dim We can approximate u

5. Galerkin Methods Linear system 1)Linear system of equation 2)square 3)Symmetric (why) 4)Positive definite Unique sol?

6. Finite Element Methods why No of elements = 16 No of nodes = 13 No interior nodes = 5 No of boundary nodes = 8 Triangulation

7. Finite Element Methods 1D Problem 0 1 0.250.5 0.75 0 1 0.250.5 0.75 0 1 0.250.5 0.75 1 1 1

8. Global basis functions

9. 1 2 3 4 5 6 7 8 9 10 11 12 13 Element Labeling 14 15 16

10. Node Labeling (global labeling) 12 34 5 6 7 8 9 1011 1213

11. global basis functions 12 34 5 6 7 8 9 1011 1213

12. 12 34 5 6 7 8 9 1011 1213 global basis functions

13. Global basis functions

14. global basis functions 12 34 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 0 1011 5 1312

15. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

16. global basis functions 12 34 6 7 8 9 0 0 0 0 0 0 0 0 1011 5 1312 0 0

17. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 12 34 5 6 7 8 9 1011 1213 Assemble linear system

18. 12 34 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 0 1011 5 1312 12 34 6 7 8 9 0 0 0 0 0 0 0 0 1011 5 1312 0 0

19. Finite Element Methods HomeWork: Compute the matrix A and the vector b then solve the linear system and write the solution as a linear combination of the basis then approximate the value of the function at (x,y)= (0.3,0.3) and (0.7,0.7). can you find the analytic solution of the problem? where f(x)= x(x-1)y(y-1) with pcw-linear

20. Approximation of u 01 02 03 04 0.0695 06 07 08 09 0.04910 0.04911 0.04912 0.04913

21. X-coordinate and y-coordinate Matrix p(2,#elements) 13121110987654321 0.750.25 0.7510.50 1001x 0.25 0.75 0.50 1 0011y Matlab matrices (computation info)

22. Boundary node vector e(#boundary node) e8e7e6e5e4e3e2e1 98764321start 14329876end Matlab matrices (computation info)

23. Node Label (local labeling) 12 3 Each triangle has 3 nodes. Label them locally inside the triangle Matlab matrices (computation info)

24. Node and Element Label Matlab matrices (computation info)

25. Local label.vs. global label Matrix t(3,#elements) 16151413121110987654321 131211105555432132141 1211101312111013 12111087692 8769131211108769121110133 Matlab matrices (computation info)