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This paper introduces general methodologies for constructing closed-formsolutions 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 ofnumerical techniques for evaluating volume integral operators and also havepotential applications in certain kinds of Trefftz finite element methods. Theequations 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 Maxwellequations. Several solutions to complex PDE systems are obtained by a potentialrepresentation and rely on the Helmholtz or Poisson solvers. Some of the casesaddressed, namely Stokes flow, Maxwell's equations and linearized Navier-Stokesequations, naturally incorporate divergence constraints on the solution. Thisarticle provides a generic pattern whereby solutions are constructed byleveraging solutions of the lowest-order part of the partial differentialoperator (PDO). With the exception of anisotropic material tensors, no matrixinversion or linear system solution is required to compute the solutions. Thiswork is accompanied by a freely-available Julia library,\texttt{ElementaryPDESolutions.jl}, which implements the proposed methodologyin an efficient and user-friendly format.

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