Accurate Implementation of the Schwarz-Christoffel Tranformation

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Accurate Implementation of the Schwarz-Christoffel
Tranformation

• Evan Warner

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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

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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)?

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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)?

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What have I written?

• I have also written a GUI with two graphs, the w
  and z planes, that display the transformation

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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

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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 lt x1 lt x2 lt x3 lt ... lt xn-1
• Testing, testing, testing
• Error bounds
• Time bounds
• How they vary with number of vertexes and aspect
  ratio of polygon