Mass-Lumped Virtual Element Method with Strong Stability-Preserving Runge-Kutta Time Stepping for Two-Dimensional Parabolic Problems

TL;DR

The paper develops a mass-lumped virtual element method (VEM) for 2D parabolic diffusion on general polygonal meshes and couples it with explicit strong stability-preserving Runge-Kutta (SSP-RK) time stepping. A diagonal lumped mass matrix is constructed by row-sum with floored weights to guarantee positivity, while stabilization vanishes on row-sums, producing an $L^2$-equivalent SPD inner product and enabling efficient explicit updates. A mesh-robust spectral bound $\lambda_{\max}((\hat{M}_h)^{-1}K_h) \le \frac{C_{\mathrm{inv}}^2}{\hat{\beta}_*} h^{-2}$ yields the diffusion CFL $\Delta t=\mathcal{O}(h^{2})$, and SSP-RK schemes preserve this stability under $\Delta t \le C_{\mathrm{SSP}}\Delta t_{\mathrm{FE}}$. Numerical experiments on distorted quadrilaterals, serendipity elements, and Voronoi polygons show optimal spatial convergence for $k=1$ ($\mathcal{O}(h)$ in $H^1$ and $\mathcal{O}(h^{2})$ in $L^2$) with no degradation from mass lumping, and confirm stability of the SSP time integrators under the predicted time-stepping scaling. The framework promises efficient explicit diffusion solvers on complex polytopal grids and extends naturally to variable coefficients, nonlinear terms, and multi-physics applications.

Abstract

This paper presents a mass-lumped Virtual Element Method (VEM) with explicit Strong Stability-Preserving Runge-Kutta (SSP-RK) time integration for two-dimensional parabolic problems on general polygonal meshes. A diagonal mass matrix is constructed via row-sum operations combined with flooring to ensure uniform positivity. Stabilization terms vanish identically under row summation, so the lumped weights derive solely from the $L^2$ projector and are computable through a small polynomial system at cost $\mathcal{O}(N_k^3)$ per element. The resulting lumped bilinear form satisfies $L^2$-equivalence with edge-count-independent constants, yielding a symmetric positive definite discrete inner product. A mesh-robust spectral bound $λ_{\max}\big((\hat{\mathbf{M}}_h)^{-1}\mathbf{K}_h\big) \le C_{\mathrm{inv}}^2/\hatβ_* \cdot h^{-2}$ is established with constants depending only on spatial dimension, polynomial order, and mesh regularity. This delivers the classical diffusion-type CFL condition $Δt=\mathcal{O}(h^2)$ for forward Euler stability and extends to higher-order SSP-RK schemes, guaranteeing preservation of energy decay, positivity, and discrete maximum principles. Numerical experiments on distorted quadrilaterals, serendipity elements, and Voronoi polygons validate the theoretical predictions: the lumped VEM with $k=1$ achieves optimal convergence rates ($\mathcal{O}(h)$ in $H^1$, $\mathcal{O}(h^2)$ in $L^2$) with no degradation from geometric distortion or mass lumping, while SSP-RK integrators remain stable under the predicted $Δt\propto h^{2}$ scaling

Figures (9)

• Figure 1: Distorted Q4 mesh (12$\times$12), $h_{\max}=0.125$.
• Figure 2: Serendipity Q8 mesh (12$\times$12), $h_{\max}=0.125$.
• Figure 3: Voronoi polygonal mesh (avg. 5.5 vertices/element).
• Figure 4: $H^1$-seminorm error convergence for distorted Q4 elements comparing RK3 and RK4 time integration.
• Figure 5: $L^2$-norm error convergence for distorted Q4 elements comparing RK3 and RK4 time integration.

Theorems & Definitions (23)

• Definition 2.1
• Remark 2.1
• Lemma 2.1
• proof
• Remark 3.1
• Lemma 3.1: Positive definite
• proof
• Lemma 3.2
• proof
• Theorem 4.1: Homogeneous case