The use of nonlinear PDEs has led to significant advancements in various fields, such as physics, biology, ecology, and quantum mechanics. However, finding multiple solutions for nonlinear PDEs can be a challenging task, especially when suitable initial guesses are difficult to obtain. In this paper, we introduce a novel approach called the Companion-Based Multilevel finite element method (CBMFEM), which can efficiently and accurately generate multiple initial guesses for solving nonlinear elliptic semi-linear equations with polynomial nonlinear terms using finite element methods with conforming elements. We provide a theoretical analysis of the error estimate of finite element methods using an appropriate notion of isolated solutions, for the nonlinear elliptic equation with multiple solutions and present numerical results obtained using CBMFEM which are consistent with the theoretical analysis.

翻译：非线性偏微分方程在物理、生物学、生态学和量子力学等多个领域取得了重要进展。然而，寻找非线性偏微分方程的多重解是一项具有挑战性的任务，尤其是在难以获得合适初值的情况下。本文提出了一种名为"伴随型多水平有限元法"（CBMFEM）的新方法，该方法能够高效且精确地生成多个初始猜测值，用于采用协调有限元方法求解具有多项式非线性项的非线性椭圆半线性方程。我们基于适当的孤立解概念，对具有多重解的非线性椭圆方程的有限元方法误差估计进行了理论分析，并展示了使用CBMFEM获得的数值结果，这些结果与理论分析一致。