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

TL;DR

This paper presents a general method for constructing polynomial solutions to linear constant-coefficient PDEs, applicable to various equations like Poisson, Helmholtz, and Maxwell, with an accompanying Julia library.

Contribution

It introduces a unified approach to generate polynomial particular solutions for many PDEs without matrix inversion, and provides a practical Julia implementation.

Findings

Constructed solutions for complex PDE systems using potential representations.

Method applies to equations with divergence constraints, like Stokes and Maxwell.

Provides an open-source Julia library for efficient solution generation.

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, \texttt{ElementaryPDESolutions.jl}, which implements the proposed methodology in an efficient and user-friendly format.