Nonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes
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The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution u∈ V:=H^2_0(Ω) to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces V_h(P) and a smoother allows rough source terms F∈ V^*=H^-2(Ω). The a priori and a posteriori error analysis in this paper circumvents any trace of second derivatives by some computable conforming companion operator J:V_h→ V from the nonconforming virtual element space V_h. The operator J is a right-inverse of the interpolation operator and leads to optimal error estimates in piecewise Sobolev norms without any additional regularity assumptions on u∈ V. As a smoother the companion operator modifies the discrete right-hand side and then allows a quasi-best approximation. An explicit residual-based a posteriori error estimator is reliable and efficient up to data oscillations. Numerical examples display the predicted empirical convergence rates for uniform and optimal convergence rates for adaptive mesh-refinement.
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