The Direct Boundary Integral Method for Laplace’s Equation

The direct boundary integral method for problems governed by Laplace’s equation is based on Green’s representation formula, which gives the potential at a point inside a region V, but not on its boundary S, in terms of single and double layer potentials with density functions equal to (a) the normal derivative of the potential on S and (b) the potential on the same boundary. For two dimensions, this formula can be discretized by dividing the boundary S into N straight line segments—boundary elements—joined end to end and assuming that both the normal derivative of the potential and the potential itself are piecewise constant over the elements. For a well-posed boundary value problem either the potential or its normal derivative will be known at the midpoint of each element, leading to a system of N simultaneous linear algebraic equations in N unknowns. In addition to explaining computational details of the basic method for constant strength elements, extensions are discussed for piecewise linear approximations of the boundary parameters, piecewise homogeneous regions, and anisotropic materials.