An Improved Adaptive Orthogonal Basis Deflation Method for Multiple Solutions with Applications to Nonlinear Elliptic Equations in Varying Domains

TL;DR

The study tackles the challenge of discovering and characterizing multiple solutions to nonlinear elliptic equations and how domain geometry influences these solutions. It introduces the improved adaptive orthogonal basis deflation method (IAOBDM), which integrates an adaptive orthogonal basis framework with a bisection–deflation scheme and a trust-region nonlinear-algebraic-solver, discretized via a Legendre–Fourier spectral method on elliptic domains. The approach extends to general (nonpolynomial) nonlinearities and is validated through numerical experiments on sine-Gordon and Ginzburg–Landau-type problems, demonstrating domain-geometry–induced changes in energy and stability, as well as delta-like concentration in the singular limit. The work provides a practical, robust tool for exploring nonconvex PDE solution landscapes, verifying theoretical results, and informing domain-design choices in applications such as topology optimization and materials science.

Abstract

Multiple solutions are common in various non-convex problems arising from industrial and scientific computing. Nonetheless, understanding the nontrivial solutions' qualitative properties seems limited, partially due to the lack of efficient and reliable numerical methods. In this paper, we design a dedicated numerical method to explore these nontrivial solutions further. We first design an improved adaptive orthogonal basis deflation method by combining the adaptive orthogonal basis method with a bisection-deflation algorithm. We then apply the proposed new method to study the impact of domain changes on multiple solutions of certain nonlinear elliptic equations. When the domain varies from a circular disk to an elliptical disk, the corresponding functional value changes dramatically for some particular solutions, which indicates that these nontrivial solutions in the circular domain may become unstable in the elliptical domain. Moreover, several theoretical results on multiple solutions in existing literature are verified. For the nonlinear Sine-Gordon equation with parameter $λ$, nontrivial solutions are found for $λ> λ_2$, here $λ_2$ is the second eigenvalue of the corresponding linear eigenvalue problem. For the singularly perturbed Ginzburg-Landau equation, highly concentrated solutions are numerically found which suggests that their convergent limit is a delta function when the perturbation parameter goes to zero

Figures (14)

• Figure 3.1: Dogleg approximation.
• Figure 4.1: Different regions with varying $b$.
• Figure 4.2: Multiple solutions of \ref{['2024081801']}-\ref{['eq:DirichletBC']} with $\lambda=30$.
• Figure 4.3: The type-I solution to \ref{['2024081801']}-\ref{['eq:DirichletBC']} with varying $b$.
• Figure 4.4: The type-III solution to \ref{['2024081801']}-\ref{['eq:DirichletBC']} with varying $b$.