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The complex variable boundary element method or CVBEM is a numerical technique that can provide solutions to potential value problems in two or more dimensions by the use of an approximation function that is derived from the Cauchy integral equation in complex analysis. Given the potential values (i.e. a Dirichlet problem) along the boundary, the typical problem is to use the potential function to solve the governing Laplace equation. In this approach, it is not necessary to know the streamline values on the boundary. The modeling approach can be extended to problems where the streamline function is needed because there are known streamline values along the problem boundary (i.e. a mixed boundary value problem). Two common problems that have such conditions are insulation on a boundary and fluid flow around a solid obstacle. In this paper, five advances in the CVBEM are made with respect to the modeling of the mixed boundary value problem; namely (1) the use of Mathematica and Matlab in tandem to calculate and plot the flow net of a boundary value problem. (2) The magnitude of the size of the problem domain is extended. (3) The modeling results include direct computation and development of a flow net. (4) The graphical displays of the total flownet are developed simultaneously. And (5) the nodal point location as an additional degree of freedom in the CVBEM modeling approach is extended to mixed boundaries. A demonstration problem of fluid flow is included to illustrate the flownet development capability.

Solving potential problems, such as those that occur in the analysis of steady-state heat transfer, electrostatics, ideal fluid flow, and groundwater flow, is important in several fields of engineering, science, and applied mathematics. Numerical solution of the relevant governing equations typically involves using techniques such as domain methods (including finite element, finite difference, or finite volume), or boundary element methods (using either real or complex variables). In this paper, the Complex Variable Boundary Element method ("CVBEM") is examined with respect to the use of different types of basis functions in the CVBEM approximation function. Four basis function families are assessed in their solution success in modeling an important benchmark problem in ideal fluid flow; namely, flow around a 90 degree bend. Identical problem domains are used in the examination, and identical degrees of freedom are used in the CVBEM approximation functions. Further, a new computational modeling error is defined and used to compare the results herein; specifically, M = E / N where M is the proposed computational error measure, E is the maximum difference (in absolute value) between approximation and boundary condition value, and N is the number of degrees of freedom used in the approximation.

a b s t r a c tThe 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.Published by Elsevier Ltd.