1. Accurate Implementation of the Schwarz-Christoffel Tranformation
Evan Warner
Accurate Implementation of the Schwarz-Christoffel Tranformation

2. What is it?
A conformal mapping (preserves angles and infinitesimal shapes) that maps polygons onto a simpler domain in the complex plane
Amazing Riemann Mapping theorem:
A conformal (analytic and bijective) map always exists for a simply connected domain to the unit circle, but it doesn't say how to find it
Schwarz-Christoffel formula is a way to take a certain subset of simply connected domains (polygons) to find the necessary mapping

3. Why does anyone care?
Physical problems: Laplace's equation, Poisson's equation, the heat equation, fluid flow and others on polygonal domains
To solve such a problem:
State problem in original domain
Find Schwarz-Christoffel mapping to simpler domain
Transform differential equation under mapping
Solve
Map back to original domain using inverse transformation (relatively easy to find)

4. Who has already done this?
Numerical methods, mostly in FORTRAN, have existed for a few decades
Various programs use various starting domains, optimizations for various polygon shapes
Long, skinny polygons notoriously difficult, large condition numbers in parameter problem
Continuous Schwarz-Christoffel problem, involving integral equation instead of discrete points, has not been successfully implemented

5. How to find a transformation...
State the domain, find the angles of the polygon, and come up with the function given by the formula:
B and A are constants determined by the solution to the parameter problem, the x's are the points of the original domains, the alphas are the angles

6. How to find a transformation...
Need a really fast, accurate method of computing that integral (need numerical methods) many many times.
Gauss-Jacobi quadrature provides the answer: quadrature routine optimized for the necessary weighting function.
Necessary to derive formulae for transferring the idea to the complex domain.

7. How to find a transformation...
The parameter problem must be solved – either of two forms, constrained linear equations or unconstrained nonlinear equations (due to Trefethen)
Solve for prevertices - points along simple domain that map to verticies
Once prevertices are found, transformation is found

8. Examples
Upper half-plane to semi-infinite strip; lines are Re(z)=constant and Im(z)=constant

9. Examples
Mapping from upper half-plane to unit square; lines are constant for the opposite image

10. What have I done so far? Implementation of complex numbers in java
ComplexFunction class
Implementation of Gauss-Jacobi quadrature
Basic graphical user interface with capability to calculate Gauss-Jacobi integrals
Testing done mostly in MATLAB (quad routine)

12. What's next?
Research into solving the nonlinear system parameter problem – compare numerical methods
Independent testing program for a variety of domains, keeping track of mathematically computed maximum error bounds
User-friendly GUI for aids in solving physical problems and equations