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The lowest-order nonconforming virtual element extends the Morley triangularelement to polygons for the approximation of the weak solution $u\inV:=H^2_0(\Omega)$ to the biharmonic equation. The abstract framework allows(even a mixture of) two examples of the local discrete spaces $V_h(P)$ and asmoother allows rough source terms $F\in V^*=H^{-2}(\Omega)$. The a priori anda posteriori error analysis in this paper circumvents any trace of secondderivatives by some computable conforming companion operator $J:V_h\to V$ fromthe nonconforming virtual element space $V_h$. The operator $J$ is aright-inverse of the interpolation operator and leads to optimal errorestimates in piecewise Sobolev norms without any additional regularityassumptions on $u\in V$. As a smoother the companion operator modifies thediscrete right-hand side and then allows a quasi-best approximation. Anexplicit residual-based a posteriori error estimator is reliable and efficientup to data oscillations. Numerical examples display the predicted empiricalconvergence rates for uniform and optimal convergence rates for adaptivemesh-refinement.

Nonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes