Readily computable Green’s and Neumann functions for symmetry-preserving triangles

Neumann and Green’s functions of the Laplacian operator on $30$-$60$-$90^{\circ }$ and $45$-$45$-$90^{\circ }$ triangles can be generated with appropriately placed multiple sources/sinks in a rectangular domain. Highly accurate and easily computable Neumann and Green’s function formulas already exist for rectangles. The extension to equilateral triangles is illustrated. In applications, closed-form expressions can be constructed for the potential, the streamfunction, or the various spatial derivatives of these properties. The derivation of analytic line integrals of these functions allows the proper handling of singularities and facilitates extended applications to problems on domains with open boundaries. Using a boundary integral method, it is demonstrated how one can construct semi-analytical solutions to problems defined on domains that exhibit spatially-dependent properties (heterogeneous media) or possess irregular boundaries.