Least squares spectral element formulation of eigenvalue problems with/without interface : the one dimensional example

TL;DR

This paper develops a least-squares based spectral-element method for one-dimensional eigenvalue problems with and without interface conditions. By weakly enforcing interface matching through a least-squares functional, it derives a discrete generalized eigenvalue problem $a(u,v)=\lambda b(u,v)$ preconditioned by an equivalent quadratic form and solved efficiently via MATLAB's eigs. The authors establish a rigorous convergence analysis for both non-interface and interface cases, showing algebraic to exponential convergence rates depending on coefficient regularity, and validate the approach with numerical experiments across multiple jump scenarios. The framework offers a robust, high-order, nonconforming discretization strategy for eigenproblems with interfaces and lays the groundwork for extension to higher dimensions.

Abstract

Here, we present a least-squares based spectral element formulation for one-dimensional eigenvalue problems with interface conditions. First we develop the method for without interface case, then we extend it to interface case. Convergence analysis for eigenvalues and eigenfunctions have been discussed. Numerical experiments with different jump conditions have been displayed.

Figures (5)

• Figure 1: One dimensional interface problem
• Figure 2: Discretization of domain $\Omega$
• Figure 3: Domain with interface discretization
• Figure 4: One dimensional interface problem
• Figure 5: $H^{1}$ error of eigenfunctions

Theorems & Definitions (9)

• Lemma 3.1
• Theorem 3.1
• Remark 3.1
• Lemma 3.2
• Theorem 3.2
• Theorem 5.1
• Remark 5.1
• Theorem 5.2
• Remark 5.2