A numerical approach for investigating multiple eigenpairs of a quasilinear elliptic system

The aim of this study is to provide a rigorous description of a numerical approach based on the local min-orthogonal method for finding multiple eigenpairs of a quasilinear elliptic system. By a Rayleigh quotient formulation, the eigenvalue problem is transformed into a constrained variational problem. In this constrained problem setup, we define a novel local L- \(\perp \) $\perp$ selection mapping on a product space of two Banach spaces in order to develop a local characterization of the eigenfunctions. Based on this local characterization, we present a numerical algorithm for computing multiple eigenpairs of the quasilinear elliptic system. For practical implementation of the proposed algorithm, we discretized the eigenvalue problem using the finite element method. Then, a global sequence convergence result of the algorithm is established for the discretized problem. Finally, the algorithm and its associated theory are demonstrated by numerical experiments.