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.

翻译：本文提出了在二维和三维空间中构造具有多项式右端项的线性常系数偏微分方程（PDEs）闭式解的通用方法。多项式解近来在体积积分算子数值技术发展及特定Trefftz有限元方法中重新获得重要应用。本研究涵盖的方程包括各向同性及各向异性Poisson方程、Helmholtz方程、Stokes方程、线性化Navier-Stokes方程、稳态对流扩散方程、弹性静力学方程，以及时谐弹性动力学方程和Maxwell方程组。复杂偏微分方程系统的多个解通过势表示获得，并依赖Helmholtz或Poisson求解器。部分讨论情形（如Stokes流、Maxwell方程组及线性化Navier-Stokes方程）天然包含对解的散度约束条件。本文提供通用模式：通过利用偏微分算子最低阶部分的解来构造解。除各向异性材料张量外，计算过程中无需矩阵求逆或线性系统求解。本研究附带开源Julia库\texttt{ElementaryPDESolutions.jl}，以高效且用户友好的格式实现了所提出的方法。