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Companion-Based Multi-Level Finite Element Method for Computing Multiple Solutions of Nonlinear Differential Equations

Wenrui Hao, Sun Lee, Young Ju Lee

TL;DR

The paper introduces CBMFEM, a companion-based multilevel finite element framework designed to compute multiple solutions of nonlinear elliptic semilinear PDEs with polynomial nonlinearities. By solving on a coarse grid and constructing a companion matrix to generate multiple informed initial guesses for finer grids, CBMFEM enables efficient discovery of distinct stationary states via Newton refinement. The authors develop an isolated-solution based a priori error analysis, establish a discrete inf-sup condition, and demonstrate convergence properties of the FEM discretization. Numerical experiments in 1D and 2D across diverse models (including Schnakenberg and Gray–Scott) validate the method's ability to uncover multiple solutions with competitive computational performance, and the work points to practical applicability in nonlinear PDEs with polynomial nonlinearities.

Abstract

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.

Companion-Based Multi-Level Finite Element Method for Computing Multiple Solutions of Nonlinear Differential Equations

TL;DR

The paper introduces CBMFEM, a companion-based multilevel finite element framework designed to compute multiple solutions of nonlinear elliptic semilinear PDEs with polynomial nonlinearities. By solving on a coarse grid and constructing a companion matrix to generate multiple informed initial guesses for finer grids, CBMFEM enables efficient discovery of distinct stationary states via Newton refinement. The authors develop an isolated-solution based a priori error analysis, establish a discrete inf-sup condition, and demonstrate convergence properties of the FEM discretization. Numerical experiments in 1D and 2D across diverse models (including Schnakenberg and Gray–Scott) validate the method's ability to uncover multiple solutions with competitive computational performance, and the work points to practical applicability in nonlinear PDEs with polynomial nonlinearities.

Abstract

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.
Paper Structure (16 sections, 5 theorems, 54 equations, 13 figures, 9 tables, 1 algorithm)

This paper contains 16 sections, 5 theorems, 54 equations, 13 figures, 9 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Governing Equations
  3. Finite element formulation and a priori error analysis
  4. A priori error analysis
  5. Companion-Based Multilevel finite element method (CBMFEM)
  6. Numerical Examples
  7. Examples for 1D
  8. Example 1
  9. Example 2
  10. Example 3
  11. Example 4
  12. The Schnakenberg model
  13. Examples in 2D
  14. Example 1
  15. The Gray–Scott model in 2D
  16. ...and 1 more sections

Key Result

lemma thmcounterlemma

Under the assumption that $\|u_h\|_{0,\infty}, \|u\|_{0,\infty} \leq \Gamma$, we have where $C$ is a constant that depends on $\Gamma$.

Figures (13)

  • Figure 1: Mesh refinement of CBMFEM in 1D (left) and 2D with edge (right). The square dots are the coarse nodes while filled circles are newly introduced fine nodes.
  • Figure 2: A flowchart of the CBMFEM for solving the nonlinear differential equation. The hierarchical structure of CBMFEM is illustrated on the left, where for each level, we obtain a solution on the coarse grid, $V_H$. We then solve local nonlinear equations by constructing companion matrices and generate initial guesses for Newton's method on the finer level $V_h$ on the right.
  • Figure 3: Numerical solutions of Eq. (\ref{['ex2']}) with $N=1025$ grid points.
  • Figure 4: Numerical solutions of Eq. (\ref{['example2']}) with $N=1025$ grid points.
  • Figure 5: Numerical solutions of Eq. (\ref{['ex3']}) with $1025$ grid points for $p=1$, $p=7$, and $p=18$, respectively.
  • ...and 8 more figures

Theorems & Definitions (7)

  • remark thmcounterremark
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • theorem 1