The weak Galerkin finite element method for the Steklov eigenvalue problem

TL;DR

This work develops a weak Galerkin finite element framework for the Steklov eigenvalue problem, enabling high-order asymptotic lower bounds and guaranteed lower bounds for eigenvalues. By leveraging nonconforming WG spaces, weak gradients, and a tunable stabilization $\gamma(h)$, the authors obtain rigorous $H^1$ and $L^2$ error estimates and show that the discrete eigenvalues bound the exact spectrum from below under suitable mesh- and parameter choices. The paper also provides error analysis for the associated source problem and introduces GLB conditions, validated by numerical experiments on square and L-shaped domains. Overall, the approach yields accurate, provably lower-bounding eigenvalue approximations on polygonal/polyhedral meshes, with potential extensions to two-grid and two-space variants for efficiency.

Abstract

This paper introduces the application of the weak Galerkin (WG) finite element method to solve the Steklov eigenvalue problem, focusing on obtaining lower bounds of the eigenvalues. The noncomforming finite element space of the weak Galerkin finite element method is the key to obtain lower bounds of the eigenvalues. The arbitary high order lower bound estimates are given and the guaranteed lower bounds of the eigenvalues are also discussed. Numerical results demonstrate the accuracy and lower bound property of the numerical scheme.

Figures (8)

• Figure 6.1: $\Omega=(0,1)^2$, $k=1$, $\gamma(h)=h^{0.1}$, $h=1/16$, the first eigenfunction.
• Figure 6.2: $\Omega=(0,1)^2$, $k=1$, $\gamma(h)=h^{0.1}$, $h=1/16$, the second eigenfunction.
• Figure 6.3: $\Omega=(0,1)^2$, $k=1$, $\gamma(h)=h^{0.1}$, $h=1/16$, the third eigenfunction.
• Figure 6.4: $\Omega=(0,1)^2$, $k=1$, $\gamma(h)=h^{0.1}$, $h=1/16$, the fourth eigenfunction.
• Figure 6.5: $\Omega=(0,1)\times (0,1)\setminus[\frac{1}{2},1]\times [\frac{1}{2},1]$, $k=1$, $\gamma(h)=h^{0.1}$, $h=1/16$, the first eigenfunction.

Theorems & Definitions (32)

• Definition 2.1
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• Lemma 3.5
• proof
• Lemma 3.6
• proof