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A Poisson Solver for Complex Geometries Using Second Kind Boundary Integral Equations and B-Spline Approximation

M. D. McGonagle

Mathematical Sciences, Clemson University


A new method is presented for solving the Poisson equation tex2html_wrap_inline44, over the interior of a 2-D region tex2html_wrap_inline46, with boundary tex2html_wrap_inline48 defined by straight line segments. Allowed boundary conditions on each line segment are Dirichlet (tex2html_wrap_inline50) or Neumann (tex2html_wrap_inline52), where tex2html_wrap_inline54 is the outward normal vector. The problem
is split into tex2html_wrap_inline56, and solved by finding a particular solution to the Poisson problem
with a fast solver over a bounding box tex2html_wrap_inline58, which contains tex2html_wrap_inline46, 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 tex2html_wrap_inline58. 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 tex2html_wrap_inline46 and from the boundary conditions, which may induce solutions tex2html_wrap_inline68 which are singular at isolated parts of tex2html_wrap_inline70. Techniques to deal with this difficulty are discussed, and extensions to this method and other research directions are included.

Dan Warner
Tue Feb 24 16:36:44 EST 1998