Poisson tau-method solvability conjecture on hypercubes

Let $\Pi_N$ denote the polynomials of degree at most $N$, and let $\Pi_N^d$ be the corresponding polynomial space in $d$ variables. Consider the Poisson equation

$$\Delta u=f$$

in a $d$-dimensional hypercube with continuous Dirichlet boundary conditions. Define

$$\Pi_{N-2,N}^{(d)}=\Pi_N^d/\Pi_{N-2}^d.$$

Poisson tau-method solvability conjecture. If $f\in\Pi_N^d$ and the tau polynomials $P_k$ span $\Pi_{N-2,N}^{(d)}$, then the tau-modified equation

$$\Delta u+\sum_k\tau_kP_k=f$$

has a unique solution $u\in\Pi_N^d$.

This conjecture proposes the multidimensional analogue of the one-dimensional sufficient condition for solvability of tau-modified differential equations. It motivates the choice of interior tau polynomials in the two-dimensional Poisson discretization, while the general solvability theory for higher-dimensional problems remains to be established.