Solving the Porous Medium Equation with the eXtreme Mesh deformation approach (X-Mesh)

TL;DR

This work addresses numerical challenges in solving the nonlinear Porous Medium Equation with sharp interfaces, where standard FEM struggles to maintain non-negativity, mass conservation, and accurate interface localization. It introduces the eXtreme Mesh deformation (X-Mesh) approach, an ALE-based moving-mesh method that preserves topology and ensures the interface is fully represented by mesh vertices, enabling robust handling of waiting times and topology changes. The method combines a Newton-Raphson solve with Lagrange multipliers to maintain non-negativity, an interface-localization strategy, and a carefully designed mesh velocity to avoid unphysical spreading, achieving improved interface localization and comparable or better convergence compared to classical FEM. Numerical results on Barenblatt-Pattle benchmarks, waiting-time tests, and topology-change scenarios demonstrate accurate interface tracking, mass conservation, and resilience to dynamic interface changes, highlighting the practical potential of X-Mesh for PME simulations.

Abstract

We introduce a new scheme for solving the non-regularized Porous Medium Equation. It is mass conserving and uses only positive unknown values. To address these typically conflicting features, we employ the eXtreme Mesh deformation approach (X-Mesh), specifically designed for problems involving sharp interfaces. The method ensures that the interface is always meshed, even in the face of complex topological changes, without the need for remeshing or altering the mesh topology. We illustrate the effectiveness of the approach through various numerical experiments.

Figures (20)

• Figure 1: Green areas correspond to the phase$\mathcal{P}_{u(t)}$ (part of the domain where $u(t) > 0$). The grey area corresponds to the empty region$\mathcal{Q}_{u(t)}$ (part of the domain where $u(t) = 0$) and red lines to the interface$\Gamma(u(t))$.
• Figure 2: Numerical solution of the PME obtained with a classical FEM. Colored areas correspond to the part of the domain $\Omega$ where $u\geq 0$. Left: initial condition at $t=t_0$. Right: numerical solution at $t_1 = t_0 + \Delta t$.
• Figure 3: Evolution of $(U_{n+1}^k, \mathbf{X}_{n+1}^k)$ when applying Algorithm \ref{['alg:xm']}. Colored areas correspond to part of the domain where $u^h_{n+1}>0$. On the left is represented the solution $U^0_{n+1}$ obtained on the mesh intial guess $\mathbf{X}_{n+1}^0 = \mathbf{X}_0$. Middle left and right present the mesh $\mathbf{X}^k_{n+1}$ and associated solution $U^k_{n+1}$ for $k=1$ and $k=2$ respectively. On the right is depicted the converged solution $(U_{n+1}, \mathbf{X}_{n+1})$.
• Figure 4: Numerical solutions obtained at $t_{n+1}$ when $\mathcal{T}_{n+1}$ does not fully describe $\Gamma^h_{n+1}$. Colored areas correspond to the part of the domain $\Omega$ where $u > 0$. Left: solution obtained with classical Newton-Raphson iterations. Right: solution obtained with Newton-Raphson iterations with Lagrange multipliers described Eq. \ref{['eq:NRmod']}. The red spheres highlight nodes associated to a Lagrange multiplier.
• Figure 5: Left: in green, nodes of $\mathcal{N}_\mathcal{P}$ and in black, nodes of $\mathcal{N}_\mathcal{Q}$. Right: In red are highligted nodes selected to belong to $\mathcal{N}(\Gamma^h_{n+1})$. The blurred blue line represent approximate location of $\Gamma^h_{n+1}$.