Work

The Dirichlet problem for Poisson’s equation in a right-isosceles triangle

Year: 2026

Type: article

Abstract: We derive an explicit series representation for the Dirichlet problem for Poisson’s equation in a right-isosceles triangle. Linearity is used to decompose the boundary data into three harmonic subproblems, each carrying the prescribed values on a single side, and a fourth Poisson subproblem with homogeneous Dirichlet conditions. The two leg-driven harmonic fields admit simple sine-hyperbolic expansions in the original coordinates, while the hypotenuse-driven field is obtained after a 45 ∘ rotation and scaling that maps the domain to a symmetric triangle. The remaining forced problem is treated by an eigenfunction expansion of the Dirichlet Laplacian, with coefficients obtained by orthogonal projection. The resulting formula provides a convenient analytic benchmark for assessing numerical solvers on polygonal domains.

Source: Boundary Value Problems

DOI: 10.1186/s13661-026-02344-3

Authors: Juliana Zago Dias Feijó, Vinícius Coutinho da Silva, André von Borries Lopes

Institution: Universidade de Brasília

Language: English

Topic: Differential Equations and Boundary Problems

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $1,890

Funder: Fundação de Apoio à Pesquisa do Distrito Federal