Generalized Finite Element Methods

1. Generalized Finite Element Methods Strong formulation, minimization principles, and weak formulation Suvranu De

2. Last 5 classes Polynomial interpolation Approximation Least squares Moving least squares Partition of unity paradigm Wavelets

3. This class Differential equations: Strong formulation (BVP) Minimization statement Weak formulation (VBVP)

4. The Poisson equation Dirichlet problem The strong formulation (BVP) Domain Find u(x) such that subject to boundary conditions • Deformation of an elastic bar • Displacement of a string with distributed load • Steady state heat conduction in a bar

5. The Poisson equation Dirichlet problem Solution: integral representation Green’s function Continuous Symmetric Positive G(x,y) x=0.5 x=0.25 x=0.75 y 0 1 Plug in and check!!

6. The Poisson equation Dirichlet problem • Properties of solution: • The solution always exists • The solution depends continuously on the data, specifically • Given f(x) the solution is unique. • If f(x) is positive, so is u(x) • u(x) is always smoother than the data f(x) The problem is well-posed

7. Minimization Principle The Dirichlet problem Functional Define functional e.g., Elastic bar: J(w) is the total potential energy of the bar

8. Minimization Principle The Dirichlet problem Statement... Find • “arg min” argument that minimizes • “”  an element of In words: Over all functions w in X, the u that satisfies makes J(w) as small as possible

9. Minimization Principle The Dirichlet problem Statement... X = { v sufficiently smooth, v(0)=v(1)=0}

10. Minimization Principle The Dirichlet problem Proof... Let w = u + v Then

11. Minimization Principle The Dirichlet problem Proof... J(u) dJv(u) first variation > 0 for v not equal 0

12. Minimization Principle The Dirichlet problem Proof...

13. the Minimization Principle The Dirichlet problem Proof... Notice: The minimizer not only satisfies the BVP but also makes the first variation of the functional vanish

14. Weak formulation (VBVP) The Dirichlet problem Statement Find such that • Note: • Corresponding to every BVP there exists a VBVP (and vice versa). • Only few BVPs have corresponding minimization statements.

15. Weak formulation (VBVP) The Dirichlet problem Definitions Linear space, Y: A set Y is a linear(or vector) spaceif

16. Weak formulation (VBVP) The Dirichlet problem Definitions Linear forms, L(v): (linear)

17. Weak formulation (VBVP) The Dirichlet problem Definitions Bilinear forms, B(w,v): B(w,v) is linear form in w for fixed v B(w,v) is linear form in v for fixed w (bilinear)

18. Weak formulation (VBVP) The Dirichlet problem Definitions SPD bilinear forms, B(w,v): is bilinear B(w,v) = B(v,w)SPD; SPD

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

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

21. Weak formulation (VBVP) The Dirichlet problem Proper space (X) Since the bilinear form a involves only first derivatives.

22. Strong formulation The Neumann problem The BVP for given f and g

23. Minimization Principle The Neumann problem Statement Find where X = { , v(0) = 0} and Extra term taking account of the Newmann b.c.

24. Weak formulation (VBVP) The Neumann problem Statement Find such that

25. Weak formulation The Neumann problem Proof... Let us derive the VBVP from the BVP. Multiply both sides by v(x)  X and integrate over the domain remember X = { , v(0) = 0}

26. Weak formulation The Neumann problem Proof...

27. Weak formulation The Neumann problem Proof... Can you derive the VBVP from the minimization principle?

28. Weak formulation The Neumann problem Essential vs. Natural Essential boundary conditions: Imposed by X Natural boundary conditions: incorporated in J (or a,l) Here: Essential Dirichlet Natural Neumann Important theoretical and numerical ramifications

29. Strong formulation Inhomogeneous Dirichlet conditions The BVP

30. Minimization Principle Inhomogeneous Dirichlet conditions Statement Find where XD = { , v(0) = 0, v(1)=uD} X = { , v(0) = v(1) = 0} Notice that XD isnot a linear space. The difference of two members in XD is in X which is a linear space and

31. Weak formulation (VBVP) Inhomogeneous Dirichlet conditions Statement Find such that

32. Summary • The Poisson problem has a strong formulation; a minimization formulation , and a weak formulation. • The minimization/weak formulations are more general than the strong formulation in terms of regularity and admissible data. • Every strong form (BVP) has a corresponding weak form (VBVP) and vice versa. But only a limited number of BVPs have a minimization principle (only if a is SPD).

33. Summary • The weak formulation is defined by: • a space X, a bilinear from a; a linear form l. • The weak formulations identify • ESSENTIAL boundary conditions. • Dirichlet ---- reflected in X; • NATURAL boundary conditions. • Neumann ---- reflected in a,l.