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Peer-reviewedCC BYBoundary Value Problems

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

  • Juliana Zago Dias Feijó1
  • Vinícius Coutinho da Silva1
  • André von Borries Lopes1
  1. 1Departamento de Engenharia Mecânica, Faculdade de Tecnologia, Universidade de Brasília
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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.

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Cite

BibTeXRIS
BibTeX
@article{feijo2026dirichlet,
  title = {{The Dirichlet problem for Poisson’s equation in a right-isosceles triangle}},
  author = {Juliana Zago Dias Feijó and Vinícius Coutinho da Silva and André von Borries Lopes},
  journal = {Boundary Value Problems},
  year = {2026},
  month = oct,
  doi = {10.1186/s13661-026-02344-3},
  url = {https://doi.org/10.1186/s13661-026-02344-3},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - The Dirichlet problem for Poisson’s equation in a right-isosceles triangle
AU  - Juliana Zago Dias Feijó
AU  - Vinícius Coutinho da Silva
AU  - André von Borries Lopes
PY  - 2026
DA  - 2026/10/25
JO  - Boundary Value Problems
DO  - 10.1186/s13661-026-02344-3
UR  - https://doi.org/10.1186/s13661-026-02344-3
AB  - 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.
SN  - 1687-2770
SN  - 1687-2762
ER  -