Generalized Finite Element Methods

1. Generalized Finite Element Methods Approximate solution techniques: Rayleigh-Ritz and Galerkin Methods Suvranu De

2. Last class Strong formulation (BVP) Minimization statement Weak formulation (VBVP)

3. This class Approximate solution techniques: Rayleigh Ritz Method Galerkin Method Other techniques Equivalence

4. The Poisson equation Dirichlet problem The strong formulation (BVP) Domain: for given f(x)

5. Minimization Principle The Dirichlet problem Statement... Find where X = and

6. Weak formulation (VBVP) The Dirichlet problem Restatement... Let a SPD bilinear form and a linear form

7. Weak formulation (VBVP) The Dirichlet problem Restatement... Minimization Principle: Weak Statement (VBVP): Find

8. Strong form The Dirichlet problem in 2D Findu(x,y)such that W G where and Wis a domain in R2with boundaryG

9. Weak formulation The Dirichlet problem in 2D Restatement

10. XhX Approximation Basis Define a finite dimensional subspace Xh of X spanned by linearly independent functions “basis functions” i.e., any function whXh may be written as dim(ension)(Xh) = N

11. XhX Approximation Basis • These functions can be • Lagrange polynomials • Least squares • Moving least squares functions • PU functions • Wavelets • …..

12. Minimization Principle The Rayleigh-Ritz Method General approach Start with the minimization principle and pose it in the subspace Xh(APPROXIMATION)

13. Minimization Principle The Rayleigh-Ritz Method General approach Approximate solution in the subspace Xh is

14. Minimization Principle The Rayleigh-Ritz Method Geometric interpretation

15. Minimization Principle The Rayleigh-Ritz Method J|Xh... Claim: For problems of type minimization of the functional in the subspace Xh is equivalent to the following problem Find such that

16. Minimization Principle The Rayleigh-Ritz Method J|Xh... Using the property of bilinearity

17. Minimization Principle The Rayleigh-Ritz Method J|Xh... Using the property of linearity Hence

18. Minimization Principle The Rayleigh-Ritz Method J|Xh... For our Dirichlet problem Notice that Ah is SPD

19. Minimization Principle The Rayleigh-Ritz Method J|Xh...

20. Minimization Principle The Rayleigh-Ritz Method J|Xh...

21. Minimization Principle The Rayleigh-Ritz Method J|Xh... Since Ah is symmetric Similarly

22. Minimization Principle The Rayleigh-Ritz Method J|Xh... • Different bases will generate different Ah - with different bandedness, sparsity and conditioning - and hence different solution. • Since Ah is SPD for the Dirichlet problem, the solution always exists and is unique.

23. VBVP The Galerkin Method • Takes the weak form (VBVP) as the starting point. • Very general. Works for problems that does not have a minimization principle. • For problems that do have a minimization principle, the Galerkin method and the the Rayleigh-Ritz methods produce exactly the same set of discrete equations.

24. VBVP The Galerkin Method Approximation Pose the weak form: Find such that in the subspace Xh(APPROXIMATION) Find such that

25. VBVP The Galerkin Method Using the property of bilinearity and linearity as before Using symmetricity of Ah

26. VBVP The Galerkin Method Discrete equations The discrete problem Find such that Choose First row

27. VBVP The Galerkin Method Discrete equations The discrete problem Find such that Choose Second row and so on....

28. VBVP The Galerkin Method Discretized set of equations The discrete problem Find such that is equivalent to Find such that

29. VBVP The Galerkin Method Example Numerical solution Analytical solution

30. Summary • The Rayleigh-Ritz Method takes the Minimization principle as its starting point • The Galerkin Method takes the Weak form as its starting point • The Galerkin Method is much more general than the Rayleigh-Ritz Method. • For problems that permit a minimization principle, both the methods give rise to the same set of equations in the same finite dimensional subspace.