Benchmarking stabilized and self-stabilized p-virtual element methods with variable coefficients

TL;DR

This work benchmarks stabilized and self-stabilized p-version Virtual Element Methods (VEM) for Laplace, linear elasticity, and Stokes problems and introduces VC-VEM, a coefficient-aware extension that incorporates variable coefficients directly into the polynomial projections. Self-stabilized formulations achieve optimal accuracy without tuning stabilization parameters, but incur worse conditioning and higher cost due to enlarged projection spaces; stabilization remains sensitive to parameter choices, especially on curved-edge meshes. The VC-VEM framework demonstrates robustness to coefficient variability across second-order elliptic and elasticity problems, outperforming standard approaches when coefficients vary in space. Together, these results provide practical guidance for selecting stabilized, self-stabilized, or coefficient-aware VEM variants in real-world, variable-coefficient, polytopal-mesh applications, with potential impact on high-order simulations of variable stiffness panels and related engineering problems.

Abstract

Standard Virtual Element Methods (VEM) are based on polynomial projections and require a stabilization term to evaluate the contribution of the non-polynomial component of the discrete space. However, the stabilization term does not generally conform to the physics of the considered problem and a criterion for its choice is not typically supported by theoretical arguments. Hence, stabilization-free and self-stabilized formulations have been proposed. Moreover, the accuracy of VEM deteriorates in case of problems with variable coefficients, that are not usually accounted when constructing polynomial projections. The mentioned aspects might limit the use of the method when tackling real-world applications. This paper provides an in-depth numerical investigation into different stabilized and self-stabilized formulations for the p-version of VEM. The results show that self-stabilized and stabilization-free formulations achieve optimal accuracy, while suffering from worse conditioning. Moreover, a new approach for dealing with variable coefficients is introduced: the discrete space is modified so that general coefficients can be accounted when computing the polynomial projectors. Numerical results show that this new approach is more robust than the standard ones.

Figures (22)

• Figure 1: Mesh types (Control points for Bézier-edge mesh in $\Omega = \left( 0,1 \right) ^2$: $\bm{P}_0 = (0,0.5),\bm{P}_1 = (0.25,0.25),\bm{P}_2 = (0.75,0.75),\bm{P}_3 = (1,0.5)$).
• Figure 2: Laplace problem: parametric study on $\tau$ for quadrilateral mesh.
• Figure 3: Laplace problem: parametric study on $\tau$ for Voronoi mesh.
• Figure 4: Laplace problem: parametric study on $\tau$ for octagonal mesh.
• Figure 5: Laplace problem: maximum polynomial order for different tolerance values.

Theorems & Definitions (1)

• Remark 1