Construction of polynomial particular solutions of linear constant-coefficient partial differential equations

Thomas G. Anderson, Marc Bonnet*, Luiz M. Faria, Carlos Pérez-Arancibia

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)

Abstract

This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial solutions have recently regained significance in the development of numerical techniques for evaluating volume integral operators and also have potential applications in certain kinds of Trefftz finite element methods. The equations covered in this work include the isotropic and anisotropic Poisson, Helmholtz, Stokes, linearized Navier-Stokes, stationary advection-diffusion, elastostatic equations, as well as the time-harmonic elastodynamic and Maxwell equations. Several solutions to complex PDE systems are obtained by a potential representation and rely on the Helmholtz or Poisson solvers. Some of the cases addressed, namely Stokes flow, Maxwell's equations and linearized Navier-Stokes equations, naturally incorporate divergence constraints on the solution. This article provides a generic pattern whereby solutions are constructed by leveraging solutions of the lowest-order part of the partial differential operator (PDO). With the exception of anisotropic material tensors, no matrix inversion or linear system solution is required to compute the solutions. This work is accompanied by a freely-available Julia library, ElementaryPDESolutions.jl, which implements the proposed methodology in an efficient and user-friendly format.
Original languageEnglish
Pages (from-to)94-103
Number of pages10
JournalComputers & mathematics with applications
Volume162
Early online date15 Mar 2024
DOIs
Publication statusPublished - 15 May 2024

Keywords

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