Stabilization-free virtual element methods based on finite element interpolation
This work addresses the computational overhead of stabilization in VEMs for polytopal discretizations, offering simpler and more efficient schemes for complex problems.
The paper introduces a stabilization-free virtual element method (VEM) framework using finite element interpolation, eliminating stabilization terms for diffusion and reaction. Two schemes are proposed: one preserving polynomial consistency and another relaxing it for simpler construction and broader applicability, with optimal convergence rates confirmed numerically.
In this paper, we introduce a new framework for designing stabilization-free virtual element methods (VEMs) based on an finite element interpolation-based strategy, where we can simultaneously eliminate the stabilization terms in the discretizations of diffusion and reaction terms. The core idea is to construct a computable, polynomial-preserving, and norm-equivalent interpolation operator from the virtual element space to a (local) finite element space. Leveraging the properties of this operator, we design two types of stabilization-free schemes. The first scheme requires the interpolation to preserve the polynomial consistency related to the bilinear forms, thereby maintaining both consistency and stability as in the standard VEM. The second scheme relaxes this consistency requirement. While it may not satisfy the standard polynomial consistency, the second scheme retains optimal convergence with simpler construction, fewer degrees of freedom and, more importantly, applicable to more complex problems such as those involving nonlinearities or variable coefficients. We construct concrete interpolation operators for both conforming and nonconforming virtual elements in two and three dimensions. These operators are then employed to realize stabilization-free schemes for conforming and nonconforming VEMs. Numerical experiments confirm the optimal convergence rates of the proposed methods. The presented framework can be extended to design stabilization-free schemes for other polytopal discretization methods, such as the hybrid high-order method and the weak Galerkin method.