  The nonconforming virtual element method (NCVEM) for the approximation of the
weak solution to a general linear second-order non-selfadjoint indefinite
elliptic PDE in a polygonal domain is analyzed under reduced elliptic
regularity. The main tool in the a priori error analysis is the connection
between the nonconforming virtual element space and the Sobolev space
$H^1_0(\Omega)$ by a right-inverse $J$ of the interpolation operator $I_h$. The
stability of the discrete solution allows for the proof of existence of a
unique discrete solution, of a discrete inf-sup estimate and, consequently, for
optimal error estimates in the $H^1$ and $L^2$ norms. The explicit
residual-based a posteriori error estimate for the NCVEM is reliable and
efficient up to the stabilization and oscillation terms. Numerical experiments
on different types of polygonal meshes illustrate the robustness of an error
estimator and support the improved convergence rate of an adaptive
mesh-refinement in comparison to the uniform mesh-refinement.
