Adaptive finite element approximations of the first eigenpair associated with $p$-Laplacian

TL;DR

This work develops an adaptive finite element method to compute the first eigenpair of the nonlinear $p$-Laplacian under Dirichlet boundary conditions. The authors formulate a discrete minimization that yields the first eigenpair on a mesh, and they construct a residual-based a posteriori estimator to drive adaptivity. They prove convergence of the discrete eigenvalues to the continuous first eigenvalue $\lambda_1$ and that the discrete eigenfunctions converge to the eigenset $E_{\lambda_1}$ in $W^{1,p}(\Omega)$, under a sufficiently fine initial mesh; auxiliary results on the residual and limiting problems underpin the analysis. Numerical experiments in 2D and 3D on multiple domains demonstrate efficient convergence, local refinement around singularities, and expected asymptotic behavior as $p\to1^+$ and $p\to\infty$. The results show that adaptive refinement captures singularities and concentrates gradients appropriately, yielding accurate first eigenpairs with fewer degrees of freedom than uniform refinements.

Abstract

In this paper, we propose an adaptive finite element method for computing the first eigenpair of the $p$-Laplacian problem. We prove that starting from a fine initial mesh our proposed adaptive algorithm produces a sequence of discrete first eigenvalues that converges to the first eigenvalue of the continuous problem and the distance between discrete eigenfunctions and the normalized eigenfunction set corresponding to the first eigenvalue in $W^{1,p}$-norm also tends to zero. Extensive numerical examples are provided to show the effectiveness and efficiency.

Figures (14)

• Figure 1: Initial meshes ($k=0$ for $\mathcal{T}_0$) with the number of vertices being 682, 365, and 741 for Examples \ref{['example1']}-\ref{['example3']} from left to right.
• Figure 2: Initial meshes ($k=0$ for $\mathcal{T}_0$) with the number of vertices being 1331, 585, and 959 for Examples \ref{['example4']}-\ref{['example6']} from left to right.
• Figure 3: Adaptive mesh refinement level with the number of vertices over each mesh as well as final computed first eigenfunctions by adaptive refinements and references for $p=1.1, 1.5, 2$ from top to bottom in Example \ref{['example1']}.
• Figure 4: Adaptive mesh refinement level with the number of vertices over each mesh as well as final computed first eigenfunctions for $p=2.5, 3, 4$ from top to bottom in Example \ref{['example1']}.
• Figure 5: Adaptive mesh refinement level with the number of vertices over each mesh as well as final computed first eigenfunctions by adaptive refinements and references for $p=10,20,30$ from top to bottom in Example \ref{['example1']}.

Theorems & Definitions (23)

• Theorem 2.1
• proof
• Lemma 3.1
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Remark 4.1
• Lemma 4.1