The Stabilization-Free Virtual Element Discretization for Anisotropic Diffusion Problems on Polygonal Meshes

This paper introduces a stabilization-free virtual element method designed for solving heterogeneous and anisotropic diffusion equations on general polygonal meshes. The new method eliminates the need for stabilization and projection operator constraints by employing the extreme learning machines to locally approximate virtual element basis functions. In numerical experiments, the proposed method produces results identical to within roundoff error to those of the standard \(Q_1\) and \(P_1\) FEM on rectangular and triangular meshes, respectively. Numerical results demonstrate that the new method maintains optimal convergence rates for the \(L^2\) and \(H^1\) error on general polygonal meshes, even when the diffusion tensor is anisotropic, heterogeneous, or discontinuous.