A numerical solution for Laplace and Poisson's equations using geometrical transformation and graph products

In this paper a numerical method using geometrical transformation and graph product rules is developed for efficient solution of the governing differential equations in many engineering problems with arbitrary domains. Initially, a mesh free formulation for rectangular domains is developed and a full decomposition of matrix equations is achieved using graph product rules. The solution of a governing equation on an arbitrary domain is sought through a geometrical transformation from the rectangular domain into the original domain using conformal mapping. Although such transformation may change the governing equation into a more complicated differential equation, it is proven that conformal mapping preserves the Laplace and Poisson's equations which are broadly used in engineering problems. The numerical implication of the conformal mapping is the existence of a unique domain partitioning in the original domain that leads to matrix equations similar to those in rectangular domain. Such unique domain partitioning, inspired by conformal mapping, reduces the computational complexity of the problem to that in a rectangular domain. The efficiency of the proposed method is examined using various engineering examples.