Nonconforming Virtual Elements for the Biharmonic Equation with Morley Degrees of Freedom on Polygonal Meshes
Carsten Carstensen, Rekha Khot, Amiya Kumar Pani · SIAM Journal on Numerical Analysis · 2023
Abstract. The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution [Formula: see text] to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces [Formula: see text] and a smoother allows rough source terms [Formula: see text]. The a priori and a posteriori error analysis in this paper circumvents any trace of second derivatives by some computable conforming companion operator [Formula: see text] from the nonconforming virtual element space [Formula: see text]. The operator [Formula: see text] is a right-inverse of the interpolation operator and leads to optimal error estimates in piecewise Sobolev norms without any additional regularity assumptions on [Formula: see text]. 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.