A treecode algorithm for the Poisson equation in a general domain with unstructured grids

Since the seminal work in 1986, the treecode algorithm has been widely used in a variety of science and engineering problems, such as the electrostatic and magnetostatic fields calculations. With the continuous advancements of science exploration and engineering applications, efficient numerical simulations for problems defined on complex domains have become increasingly necessary. In this paper, based on a hierarchy geometry tree, an efficient implementation of the treecode algorithm is described in detail for the numerical solution of a Poisson equation defined on a general domain. The features of our algorithm include: i) with the hierarchy geometry tree, the neighbor and non-neighbor patches for a given element can be generated efficiently, ii) no restriction on the geometry of the domain, which means that our algorithm can be applied for general problem, iii) the desired computational complexity \({\varvec{\mathcal {O}}}(\varvec{N}\,\varvec{\log }\,{\varvec{N}})\) $\mathcal{O}(\mathit{N}\mathbf{log}\mathit{N})$ can be observed well, where \(\varvec{N}\) $\mathit{N}$ denotes the number of degrees of freedom in the domain, and iv) very friendly to the parallel computing, i.e., an ideal speedup can be observed successfully from numerical results with OpenMP technique. It is believed that our solution potentially is a quality candidate for implementing the treecode algorithm for problems defined on general domains with unstructured grids.