A new a posteriori functional majorant is obtained for the error of approximate solutions to an elliptic equation of order 2n, n ≥ 1, with an arbitrary nonnegative constant coefficient σ ≥ 0 in the lowest order term σu, where u is the solution of the equation. The majorant is much more accurate than Aubin’s majorant, which makes no sense at σ ≡ 0 and coarsens the error estimate for σ from a significant neighborhood of zero. The new majorant also surpasses other majorants having been obtained for the case σ ≡ 0 over recent decades. For solutions produced by the finite element method on quasi-uniform grids, it is shown that the new a posteriori majorant is sharp in order of accuracy, which coincides with that of sharp a priori error estimates.