Companion-based multi-level finite element method for computing multiple solutions of nonlinear differential equations

Abstract

The utilization of nonlinear differential equations has resulted in remarkable progress across various scientific domains, including physics, biology, ecology, and quantum mechanics. Nonetheless, obtaining multiple solutions for nonlinear differential equations can pose considerable challenges, particularly when it is difficult to find suitable initial guesses. To address this issue, we propose a pioneering approach known as the Companion-Based Multilevel Finite Element Method (CBMFEM). This novel technique efficiently and accurately generates multiple initial guesses for solving nonlinear elliptic semi-linear equations containing polynomial nonlinear terms through the use of finite element methods with conforming elements. As a theoretical foundation of CBMFEM, we present an appropriate and new concept of the isolated solution to the nonlinear elliptic equations with multiple solutions. The newly introduced concept is used to establish the inf-sup condition for the linearized equation around the isolated solution. Furthermore, it is crucially used to derive a theoretical error analysis of finite element methods for nonlinear elliptic equations with multiple solutions. A number of numerical results obtained using CBMFEM are then presented and compared with a traditional method. These not only show the CBMFEM's superiority, but also support our theoretical analysis. Additionally, these results showcase the effectiveness and potential of our proposed method in tackling the challenges associated with multiple solutions in nonlinear differential equations with different types of boundary conditions.

Keywords: Boundary conditions; Elliptic semilinear PDEs; Finite element method; Multiple solutions; Nonlinear ODEs.

Fig. 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.

Fig. 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.

Fig. 3.

Numerical solutions of Eq. (49) with $N=1025$ grid points. The unstable solution is plotted with dashed lines, while the stable solution is represented with solid lines.

Fig. 4.

Numerical solutions of Eq. (56) with $N=1025$ grid points. The unstable solution is plotted with dashed lines, while the stable solution is represented with solid lines.

Fig. 5.

Numerical solutions of Eq. (57) with 1025 grid points for $p=1,p=7$, and $p=18$, respectively. Unstable solutions are plotted with dashed lines, while stable solutions are represented with solid lines.

Fig. 6.

Numerical solutions of Eq. (58) with $N=1025$ grid points for different $r$ and $d$. We have symmetric solutions so we only concern with one of them. Unstable solutions are plotted with dashed lines, while stable solutions are represented with solid lines.

Fig. 7.

Bifurcation diagram of Eq. (58) with respect to $r$ with $N=1025$ grid points and $d=1$.

Fig. 8.

3 different solutions for equation (59) with $N=1025$ grid points. The same colors are paired solutions.

Fig. 9.

Multi-level grid based on the edge refinement with a rectangular domain.

Fig. 10.

Multiple solutions of Eq. (61) with $s=1600$ and a step size of $2^{-7}$.

Fig. 11.

Bifurcation diagram of solutions of Eq. (61) with respect to $s$.

Fig. 12.

We have 24 solutions and plot only $A(x,y)$ from (63) with a step size of $2^{-6}$. The initial guesses were refined by considering both real solutions and real parts of complex solutions on the coarsest grid $(\ell =0)$.

Fig. 13.

We have 88 solutions and plot only $A(x,y)$ from (63) with a step size of $2^{-6}$. The initial guesses were refined by considering only the real solutions on the $\ell =1$ grid.