A Moving Mesh Method for Porous Medium Equation by the Onsager Variational Principle

TL;DR

This work develops a moving-mesh finite element framework for the porous medium equation by leveraging the Onsager variational principle to preserve the energy-dissipation structure. The approach yields a mixed semi-discrete formulation and two full-discretization schemes (a decoupled explicit and a robust implicit) that advance density and mesh while maintaining stability. Numerical experiments in 1D and 2D demonstrate optimal convergence with appropriate initial meshes and automatic waiting-time capture, and illustrate the method’s capability on complex geometries. The results offer a variationally grounded, adaptive method for accurately tracking free-boundary dynamics in porous media, with potential extensions to higher dimensions and orders.

Abstract

In this paper, we introduce a new approach to solving the porous medium equation using a moving mesh finite element method that leverages the Onsager variational principle as an approximation tool. Both the continuous and discrete problems are formulated based on the Onsager principle. The energy dissipation structure is maintained in the semi-discrete and fully implicit discrete schemes. We also develop a fully decoupled explicit scheme by which only a few linear equations are solved sequentially in each time step. The numerical schemes exhibit an optimal convergence rate when the initial mesh is appropriately selected to ensure accurate approximation of the initial data. Furthermore, the method naturally captures the waiting time phenomena without requiring any manual intervention.

Figures (8)

• Figure 1: Convergence of the numerical solutions for the PME ($m = 2, 5$) at $T = 2$.
• Figure 2: The exact solutions (red solid line) and the numerical solutions (blue circles) of the PME($m=5$).
• Figure 3: Numerical solutions of the PME($m=4$) with the initial value \ref{['waiting_time']} with $\theta = 0$. Here $N = 48, \tau = 2.5\times 10^{-4}$.
• Figure 4: Boundary motion with respect to time.
• Figure 5: Convergence of the numerical solutions of the PME ($m = 2,5$) at $T = 2$.

Theorems & Definitions (12)

• Proposition 1
• proof
• Theorem 1
• proof
• Proposition 2
• proof
• Remark 1
• Theorem 2
• proof
• Remark 2