1. Generalized Finite Element Methods
Spring 2005
Integral Equation Methods
Discretization Convergence Theory
Suvranu De

2. Outline
Integral Equation Methods
Reminder about Galerkin and Collocation
Example of convergence issues in 1-D
First and second kind integral Equations
Develop some intuition about the difficulties
Convergence for second-kind equations
Show consistency and stability issues
Nystrom Methods
High Order Convergence

3. Basis Function Approach
Integral Equation Basics
Basic Idea
Integral Equation:
Example Basis
Represent circle with straight lines

4. Basis Function Approach Piecewise Constant Straight Sections Example.
Integral Equation Basics
Piecewise Constant Straight Sections Example.
1) Pick a set of n Points on the surface
2) Define a new surface by connecting points with n lines.

5. Basis Function Approach Residual Definition and minimization
Integral Equation Basics
Residual Definition and minimization

6. Basis Function Approach Residual minimization using test functions
Laplace’s Equation in 2-D
Residual minimization using test functions

7. Basis Function Approach
Laplace’s Equation in 2-D
Collocation

8. Basis Function Approach
Laplace’s Equation in 2-D
Galerkin
A is symmetric

9. The density must be computed
Example Problems
Convergence Analysis
1-D First Kind Equation
The potential is given
The density must be computed
Solution = 3x

10. Collocation Discretization of 1-D Equation
Example Problems
Convergence Analysis
Collocation Discretization of 1-D Equation
Centroid Collocated Piecewise Constant Scheme

11. Collocation Discretization of 1-D Equation - The Matrix
Example Problems
Convergence Analysis
Collocation Discretization of 1-D Equation - The Matrix
One column for each density value
One row for each collocation point

12. Answers Are Getting Worse!!!
Numerical Results with increasing n
n = 40
n = 10
n = 20
Answers Are Getting Worse!!!

13. 1-D Second Kind Equation
Example Problems
Convergence Analysis
1-D Second Kind Equation
The potential is given
The density must be computed
Solution = 3x

14. Collocation Discretization of 1-D Equation
Example Problems
Convergence Analysis
Collocation Discretization of 1-D Equation
Centroid Collocated Piecewise Constant Scheme

15. Collocation Discretization of 1-D Equation - The Matrix
Example Problems
Convergence Analysis
Collocation Discretization of 1-D Equation - The Matrix

16. Answers Are Improving!!! Numerical Results with increasing n n = 40

17. 1-D First Kind Equation Difficulty
Example Problems
Convergence Analysis
1-D First Kind Equation Difficulty
Denote the integral operator as K
The integral operator is singular -1 1

18. 1-D First Kind Difficulty from the Matrix
Example Problems
Convergence Analysis
1-D First Kind Difficulty from the Matrix
Collocation Generates a Discrete form of K
Solution = 3x

19. Eigenvalues accumulating at zero.
Numerical Results with increasing n
n = 10
n = 40
n = 20
Eigenvalues accumulating at zero.

20. Intuition about the Eigenvalues
Example Problems
Convergence Analysis
Intuition about the Eigenvalues
As the discretization is refined, K’s null space can be more accurately represented

21. 2-D Kind Equation has fewer problems
Example Problems
Convergence Analysis
2-D Kind Equation has fewer problems
The second Kind equation -1 1 -1 1 + = -1 1

22. Eigenvalues do not get closer to zero.
Numerical Results with increasing n
n = 10
n = 40
n = 20
Eigenvalues do not get closer to zero.

23. General 2nd-Kind Integral Equation
Second Kind Theory
Convergence Analysis
Basic Assumptions
General 2nd-Kind Integral Equation
Assume the equation is uniquely solvable
Then a discretization method converges if consistent

24. The point collocation equation for the ith collocation point
Second Kind Theory
Convergence Analysis
Kn for collocation.
Aim: Derive a semidiscrete form of the second kind equation
The point collocation equation for the ith collocation point
Let us denote
This is the projection of s(x) onto the basis function space

25. Convergence Analysis Second Kind Theory Hence
Kn for collocation.
Hence
Let us define the interpolation operator
for any function u(x)
Hence

26. Convergence Analysis Second Kind Theory From last slide
Kn for collocation.
From last slide
We will define the semidiscrete operator Kn such that
Hence

27. Semidiscrete Integral Equation
Second Kind Theory
Convergence Analysis
Rough Proof
Operator Form for
the integral equation
Semidiscrete Integral Equation
Subtracting
And neglecting (Y-V Y)

28. The equation for the solution error (previous slide)
Second Kind Theory
Convergence Analysis
Rough Proof continued
The equation for the solution error (previous slide)
Taking norms
Error which should go to zero as n increases
Needs a bound, this is discrete stability
Goes to zero with n by consistency

29. Deriving the stability bound
Second Kind Theory
Convergence Analysis
Stability Bound
Norm of Solution error
Deriving the stability bound
Taking Norms
Bounded by C by Assumption

30. Stability Bound Continued
Second Kind Theory
Convergence Analysis
Stability Bound Continued
Repeating from last slide
Bounded by C by Assumption
Bounding Terms
Less than for n larger than n0, by consistency

31. What does it mean? Convergence Analysis Final Result
Second Kind Theory
Convergence Analysis
Rough Proof completed
Final Result
What does it mean?
The discretization convergence of a second kind integral equation solver depends only on how well the integral is approximated.

32. Collocation Discretization of 1-D Equation
1-D Second Kind Example
Nystrom Method
Collocation Discretization of 1-D Equation
Integral Equation
Apply Quadrature to Collocation Equation
Now set quadrature points = collocation points

33. Collocation Discretization of 1-D Equation Continued
1-D Second Kind Example
Nystrom Method
Collocation Discretization of 1-D Equation Continued
Set quadrature points = collocation points
System of n equations in n unknowns
Collocation equation per quad/colloc point
Unknown density per quad/colloc point

34. 1-D Discretization - Matrix Comparison
1-D Second Kind Example
Nystrom Method
1-D Discretization - Matrix Comparison
Nystrom Matrix
Piecewise constant collocation Matrix

35. 1-D Discretization - Matrix Comparison Continued
1-D Second Kind Example
Nystrom Method
1-D Discretization - Matrix Comparison Continued
Nystrom Matrix
Just Green’s function evals - No integrals
Entries each have a quadrature weight
Collocation points are quadrature points
High order quadrature = faster convergence?
Piecewise constant collocation Matrix
Integrals of Green’s functions over line sections
Distant terms equal Green’s function
Collocation points are basis function centriods
Low order method always

36. Gauss Quad ->Exponential Convergence!
1-D Second Kind Example
Nystrom Method
Convergence Theorem
What does this mean?
The discretization error for the Nystrom method applied to a second kind integral equation converges AT THE SAME RATE as the underlying quadrature!
Gauss Quad ->Exponential Convergence!

37. Convergence Comparison
1-D Second Kind Example
Nystrom Method
Convergence Comparison
Piecewise-Constant
Centroid Collocation E r o
Gauss-Quad
Nystrom n

38. 1-D Second Kind Example
Nystrom Method
Convergence Caveat
If Nystrom methods can have exponential convergence, why use anything else?
Gaussian Quadrature has exponential convergence only for nonsingular kernels
Most physical problems of interest have singular kernels (1/r, e^ikr/r, etc)

39. Summary Integral Equation Methods Reviewed Galerkin and Collocation
Example of convergence issues in 1-D
First and second kind integral Equations
Examined by example operator null spaces
Convergence for second-kind equations
Show consistency and stability issues
Nystrom Methods
High Order Convergence
Did not touch on singular integrands, the most common case in practice.