M. D. McGonagle
Mathematical Sciences, Clemson University
A new method is presented for solving the Poisson equation 
, over the interior of a 2-D region 
, with boundary 
defined by straight line segments. Allowed boundary conditions on each line segment are Dirichlet (
) or Neumann (
), where 
is the outward normal vector.
The problem
is split into 
, and solved by finding a particular solution to the Poisson problem
with a fast solver over a bounding box 
, which contains , and then solving the Laplace problem
The Poisson solver returns a Tensor product B-spline, which allows evaluation of the solution v and its partial derivatives at any point in . The Laplace problem is solved using B-spline basis functions to numerically integrate the second kind BIEs associated with the Laplace equation. Difficulty arises from both the geometry of the region and from the boundary conditions, which may induce solutions 
which are singular at isolated parts of 
. Techniques to deal with this difficulty are discussed, and extensions to this method and other research directions are included.