The 3D Nonconforming Virtual Element Method with enhanced spaces: Numerical investigation and implementation guide

Abstract In this paper, we present the implementation details of the three-dimensional Nonconforming Virtual Element Method, building upon the theoretical framework introduced in ( ESAIM: Math. Model. Numer. Anal. , 50 (3), 879–904, (2016), ( IMA J. Numer. Anal. , 37 , (2015). In this formulation, the local spaces are suitably enhanced to allow the construction of the $$\varvec{L}^2$$ projection of virtual functions onto polynomials of degree $$\varvec{k}$$ . This higher-order projection provides a more accurate representation of the load term and, more importantly, enables the definition of the reaction term. We perform a numerical convergence study of the $$\varvec{L}^2$$ -norm and $$\varvec{H}^1$$ -seminorm errors with respect to both mesh size and polynomial degree in the three-dimensional setting. Numerical results are then compared with those obtained using the conforming VEM, showing the consistency and computational performance of the proposed 3D implementation.