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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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)

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

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

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

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.

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

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

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

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

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!

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

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)

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.