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

TL;DR

This paper introduces a nonconforming virtual element method for the biharmonic equation on polygonal meshes, extending Morley elements, with error analysis and adaptive refinement demonstrating optimal convergence.

Contribution

It develops a novel nonconforming virtual element framework for biharmonic problems, including error estimation and adaptive algorithms, without requiring second derivative traces.

Findings

Optimal error estimates achieved without second derivative traces

Reliable and efficient residual-based a posteriori error estimator

Numerical results confirm empirical and optimal convergence rates

Abstract

The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution $u\in V:=H^2_0(\Omega)$ to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces $V_h(P)$ and a smoother allows rough source terms $F\in V^*=H^{-2}(\Omega)$. The a priori and a posteriori error analysis in this paper circumvents any trace of second derivatives by some computable conforming companion operator $J:V_h\to V$ 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.