On the Generalization of Numerical Modeling for Poisson Problem in Rn

The numerical solution to the Poisson problem is widely known and studied in various fields of science for its vast applications. However, most applications consider the case in two dimensions, with fewer studies addressing the problem in dimension three or higher. Most literature texts present a two-dimensional case implementation using an approach that makes it difficult to extend to higher dimensions. Our work aims to propose a generalization of the numerical solution to the Poisson problem that can be implemented for any dimension. The strategy used considers an index function that enumerates the elements of the discretized domain so that, using this index, the implementation is easily extended to any dimension. In addition to the numerical solution extension, we developed the mathematical foundation for the consistency and stability of the solution in arbitrary dimension. The preliminary results consider the implementation in Python and experiments that demonstrate the feasibility of the proposed methodology