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. What have I written?
I have written a Newton-Raphson routine to solve the parameter problem – takes the vertexes and calculates the prevertexes by taking finding an approximate Jacobian matrix at each step and following the gradient “downhill” to the root
Requires linear equation solver at every step; I use an LU factorization
Requires an approximate Jacobian matrix at each step; I take a simple forward difference to save on function evaluations (which are very expensive)‏

5. What have I written?
I have also written a GUI with two graphs, the w and z planes, that display the transformation:

6. So why isn't it correct?
I have implemented compound Gauss-Jacobi quadrature, which recursively divides the integrals into subintervals based on whether a singularity is nearby
Unfortunately my recursion skills are not quite up to par, apparently – still doesn't work

7. What's next? Fixing compound Gauss-Jacobi quadrature
Implementation of a change in variables in the equation solver in order to preserve a strict inequality among the vertexes; that is, to ensure that x0 < x1 < x2 < x3 < ... < xn-1
Testing, testing, testing:
Error bounds
Time bounds
How they vary with number of vertexes and aspect ratio of polygon