Generalized HPC method for the Poisson equation

An efficient and innovative numerical algorithm based on the use of Harmonic Polynomials on each Cell of the computational domain (HPC method) has been recently proposed by Shao and Faltinsen (2014) [1], to solve Boundary Value Problem governed by the Laplace equation. Here, we extend the HPC method for the solution of non-homogeneous elliptic boundary value problems. The homogeneous solution, i.e. the Laplace equation, is represented through a polynomial function with harmonic polynomials while the particular solution of the Poisson equation is provided by a bi-quadratic function. This scheme has been called generalized HPC method. The present algorithm, accurate up to the 4th order, proved to be efficient, i.e. easy to be implemented and with a low computational effort, for the solution of two-dimensional elliptic boundary value problems. Furthermore, it provides an analytical representation of the solution within each computational stencil, which allows its coupling with existing numerical algorithms within an efficient domain-decomposition strategy or within an adaptive mesh refinement algorithm.