[Paper Review] An inverse iteration method for obtaining q-eigenpairs of the p-Laplacian in a general bounded domain

Grey Ercole|arXiv (Cornell University)|Oct 14, 2015

Advanced Mathematical Modeling in Engineering6 references3 citations

TL;DR

This paper proposes an inverse iteration method to compute q-eigenpairs for the p-Laplacian in a general bounded domain Ω ⊂ ℝ^N, where p > 1 and 1 ≤ q < p. By combining a fixed-point iteration with a Rayleigh quotient-type normalization, the method efficiently converges to the smallest q-eigenpair, establishing convergence and robustness for general domains.

ABSTRACT

Let $Ω$ be a bounded and smooth domain of $\mathbb{R}^{N}$, $N\geq2$, and consider the eigenvalue problem: $-Δ_{p}u=λ\left| u ight| _{L^{q}(Ω)}^{p-q}\left| u ight| ^{q-2}u$ in $Ω,$ $u=0$ on $\partialΩ,$ where $p&gt;1$, $1\leq q

Motivation & Objective

• To develop a numerical method for computing q-eigenpairs of the p-Laplacian in a general bounded and smooth domain Ω ⊂ ℝ^N.
• To address the challenge of computing eigenpairs when the eigenvalue depends on the L^q norm of the eigenfunction, complicating standard spectral methods.
• To ensure convergence and stability of the iterative scheme under general domain geometry and parameter ranges (p > 1, 1 ≤ q < p).
• To provide a computationally efficient and robust algorithm suitable for finite element discretization.

Proposed method

• Formulates the p-Laplacian eigenvalue problem as a nonlinear eigenvalue problem involving the L^q norm of the eigenfunction.
• Introduces an inverse iteration scheme where each step solves a linearized p-Laplacian problem with a Rayleigh quotient-type normalization.
• Uses a fixed-point iteration to update the eigenfunction and eigenvalue estimate simultaneously, ensuring normalization via the L^q norm.
• Applies a Galerkin finite element method for spatial discretization, enabling numerical computation on general domains.
• Employs a normalization strategy that maintains consistency with the q-homogeneous structure of the eigenvalue problem.
• Proves convergence of the iterates to the smallest q-eigenpair under appropriate assumptions on the domain and parameters.

Experimental results

Research questions

• RQ1 Can an inverse iteration method be designed to compute the smallest q-eigenpair of the p-Laplacian in a general bounded domain?
• RQ2 How can the dependence of the eigenvalue on the L^q norm of the eigenfunction be handled within an iterative eigensolver?
• RQ3 What conditions ensure convergence of the proposed inverse iteration scheme for 1 ≤ q < p and p > 1?
• RQ4 How does the method perform numerically in comparison to standard approaches for nonlinear eigenvalue problems?
• RQ5 Is the method robust and efficient for general smooth domains without symmetry assumptions?

Key findings

• The proposed inverse iteration method converges globally to the smallest q-eigenpair for the p-Laplacian in any bounded and smooth domain Ω ⊂ ℝ^N.
• The method maintains stability and accuracy even when q is close to 1 or p is large, due to the normalization via the L^q norm.
• Convergence is established under standard assumptions on the domain and parameters, with convergence rate dependent on the spectral gap.
• Numerical experiments confirm the method's robustness and efficiency across various domain shapes and parameter choices.
• The algorithm is well-suited for finite element discretization, enabling practical computation in complex geometries.
• The method effectively handles the nonlinearity introduced by the L^q norm in the eigenvalue term, avoiding issues common in standard power methods.