Galerkin Method Properties in Approximation Techniques

1. Generalized Finite Element Methods Other approximation methods Suvranu De

2. Last class Approximation properties of the Galerkin method

3. Galerkin method Last class Properties Property 1:The discretization error is orthogonal to the approximation space Xh in the energy norm Property 2: Strain energy of mathematical model Strain energy of discretized model Increased refinement

4. Galerkin method Last class Properties Property 3:Best approximation property

5. This class Other approximation techniques Point collocation method Subdomain integral/average error method Weighted residual method (GUSWF I) Petrov-Galerkin method (GSWF) H-1 method (GUSWF II)

6. Convection-Diffusion Motivation given functions of (x,y) Scalar, Linear, Parabolic equation

7. Convection-Diffusion Motivation Applications • Ifu(x,y) is.... • Temperature  Heat Transfer • Pollutant concentration  Coastal Engineering • Probability distribution  Statistical Mechanics • Price of an option  Financial Engineering • ....

8. Convection-Diffusion Motivation 1D example... 1D steady state fluid flow in a channel with heat transfer x=0 x=L v qL qR x • Notice • Elliptic • Very close to the Poisson problem except for an additional derivative of odd order.

9. Convection-Diffusion Motivation 1D example... The BVP

10. Convection-Diffusion Motivation 1D example... The VBVP

11. Convection-Diffusion Motivation 1D example... Solution using Galerkin method shows spurious oscillations

12. Model problem BVP The strong formulation (BVP) boundary Domain Find u  Xsuch that subject to boundary conditions • u may be a scalar or a vector (e.g., elasticity/fluid flow in 2/3D) • The operators A and B can be scalar/vector. • The problem may be posed in 1/2/3 D

13. XhX Approximation Residuals Define a finite dimensional subspace Xh of X spanned by linearly independent functions “trial/basis functions” Assume Domain residual Boundary residual

14. Scheme Point collocation method Make the domain and boundary residuals vanish at N points Domain residual Boundary residual

15. Example Point collocation method

16. Example Point collocation method Analytical solution

17. Example Point collocation method Numerical solution Check Satisfies the essential BC automatically (no need to consider boundary residual) Domain residual

18. Example Point collocation method Numerical solution In the point collocation method we pick any point x=x1 in the domain and set the domain residual to zero at that point Domain residual

19. Example Point collocation method Numerical solution For x1<0.5 Soln symmetric about x=0.5 The max overshoots As x1 0, (uh)max 

20. Example Point collocation method Numerical solution For x1>0.5, f(x1)=0, hence uh(x)=0 !!! Solution very sensitive to the choice of collocation point

21. Example Point collocation method Analytical solution Satisfies the essential BC automatically (no need to consider boundary residual) • What happens if • x1 and x2 are both greater than 0.5? • x1 x2 ?

22. Example Point collocation method Numerical solution For x1=0.2, x2=0.8 Right bias The max overshoots

23. Comments Point collocation method Numerical solution • Solution sensitive to “proper” choice of collocation points. • should not be too close • should be enough points on the boundary • The coefficient matrix is nonsymmetric • Simple and rapid • Domain residual may not be small in between nodal points • A smooth trial function space is required (in this example, need C1 otherwise the second derivatives would not exist) • The bandwidth of the matrix depends on the support of the approximation functions

24. assume that the satisfy all the boundary conditions W Wi Scheme Subdomain collocation method Domain residual Subdivide the domain into nonoverlapping subdomains and set the integralof the domain residual to zero over each subdomain.

25. Example Subdomain collocation method

26. Example Subdomain collocation method Numerical solution Satisfies the essential BC automatically (no need to consider boundary residual) Domain residual Integral of domain residual

27. Example Subdomain collocation method Numerical solution Soln symmetric about x=0.5 The max overshoots (not as much as point collocation).

28. Example Subdomain collocation method Analytical solution Satisfies the essential BC automatically (no need to consider boundary residual) Subdivide domain into 2 equal intervals Domain residual Set integral of domain residual to zero on each subdomain

29. Example Subdomain collocation method Numerical solution

30. Comments Subdomain collocation method Numerical solution • Behavior of numerical solution much betterthan point collocation • The coefficient matrix is nonsymmetric • More complex than point collocation since integrals have to be evaluated • A smooth trial function space is still required (in this example, need C1) • It is better to set the integral of the residual to zero than just the residual to zero at a few collocation points. • The bandwidth of the matrix depends on the size of the min(size of subdomain, support of approximation function)

31. Wi h 0 h 1 (i-1)h ih Comments Reducing the constraint Subdomain collocation method Less smooth trial functions Numerical solution where 1. N equations in Nunknowns 2. Requires the trial functions to be in C1

32. Wi h 0 h 1 (i-1)h ih Comments Reducing the constraint Subdomain collocation method Less smooth trial functions Numerical solution Integrate by parts • Same answer as before • Requires the trial functions to be in C0 ! • Starting point for “finite volume” methods (FVM)

33. n Gi Gq W Wi Gu Comments Reducing the constraint Subdomain collocation method 2/3D Numerical solution Poisson’s equation Subdomain integral over Wi Use Green’s theorem

34. Comments Reducing the constraint Subdomain collocation method 2/3D Numerical solution Needs the trial functions to be C0

35. assume that the satisfy all the boundary conditions Scheme Least squares method Domain residual Minimize the L2 norm of the residual on the domain

36. Example Least squares method

37. Example Least squares method Numerical solution Domain residual L2 norm of residual Obtain ith equation by setting

38. Example Least squares method Numerical solution Symmetric matrix

39. Example Least squares method Numerical solution Satisfies the essential BC automatically (no need to consider boundary residual)

40. Example Least squares method Numerical solution Soln symmetric about x=0.5 The max undershoots (unlike point/subdomain collocation).

41. Example Least squares method Analytical solution Satisfies the essential BC automatically (no need to consider boundary residual)

42. Example Least squares method Numerical solution Right bias

43. Comments Least squares method Numerical solution • Behavior of numerical solution much betterthan point / subdomain collocation • The coefficient matrix is symmetric and positive definite • A smooth trial function space is still required (in this example, need C1) • The bandwidth of the matrix depends on the support of the approximation functions

44. assume that the satisfy all the boundary conditions Comments Scheme Weighted residual method GUSWF I Numerical solution Domain residual Set the weighted integral of the residual to zero over the entire domain. Global unsymmetric weak form I (GUSWF I)

45. Comments Scheme Weighted residual method GUSWF I Numerical solution Choose Yh: Test function space yj(x) : Test function Xh: Trial function space jj(x) : Trial function N conditions to determine the N unknown uhjs

46. Comments Scheme Weighted residual method GUSWF I Numerical solution • Note • yi(x)= d(x-xi): POINT COLLOCATION • : SUBDOMAIN COLLOCATION

47. n Gi Gq W Wi Gu Comments Scheme Petrov-Galerkin method GUSWF I Numerical solution Start from GUSWF I with distincttrial and test function spaces e.g., Poisson’s equation GUSWF I : Find uh Xh such that

48. Comments Scheme Petrov-Galerkin method GSWF Numerical solution GUSWF I requires the trial functions to be at least C1 but the test functions can be C0 (unsymmetry!) To reduce the smoothness requirement on the trial functions by using Green’s theorem Global symmetric weak form (GSWF) Find uh Xh such that

49. Comments Scheme Petrov-Galerkin method GSWF Numerical solution Choose trial functionsjj(and thereforeuh) to satisfy the prescribed Dirichlet BC Choose the test functions to vanish on the Dirichlet boundary to get rid of the boundary integral on Gu 0 Find uh Xh such that

50. Comments Scheme Petrov-Galerkin method GSWF Numerical solution