A computational approach to determining CVBEM approximate boundaries

The complex variable boundary element method (CVBEM) provides solutions of partial differential equations of the Laplace and Poisson type. Because the CVBEM is based upon convex combinations from a basis set of functions that are analytic throughout the problem domain, boundary, and exterior of the problem domain union boundary (except along branch cuts), both the real and imaginary parts of the CVBEM approximations satisfy the Laplace equation, leaving the modeling error reduction effort to be that of fitting the problem boundary conditions. In this paper, the approximate boundary approach is used to depict the goodness of fit between the CVBEM results and the problem boundary conditions. The approximate boundary is the locus of points where the CVBEM approximation function meets the values of the problem boundary conditions. Because of the collocation method, the approximate boundary necessarily intersects the problem boundary at least at the collocation points specified on the problem boundary. Consequently, adding nodes and collocation points on the problem boundary results in reducing the departure between the approximate boundary and the true problem boundary. Thus, the approximate boundary is developed by tracking level curves from the real and/or imaginary parts of the CVBEM approximation function.