06th October – Rekha Khot: Nonconforming virtual elements for the biharmonic equation with Morley degrees of freedom on polygonal meshes

The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution $u \in  V := H_0^2(Ω)$ to the biharmonic equation. Two nonconforming virtual element spaces have been introduced in [1, 4] for $H^3$ regular solutions, while a medius analysis in [3] allows minimal regularity. In this talk, we will discuss an abstract framework (cf. [2]) with two hypotheses (H1)-(H2) for unified stability and a priori error analysis of at least two different discrete spaces (even a mixture of those). A smoother J allows rough source terms $F \in V^∗ = H^{−2}(\Omega)$. The a priori and a posteriori error analysis circumvents any trace of second derivatives by a computable conforming companion operator $J : V_h \rightarrow V$ 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 $$u \in V$$. 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.