---
abstract: 'An adaptive moving mesh finite element method is studied for the numerical solution of the porous medium equation with and without variable exponents and absorption. The method is based on the moving mesh partial differential equation approach and employs its newly developed implementation. The implementation has several improvements over the traditional one, including its explicit, compact form of the mesh velocities, ease to program, and less likelihood of producing singular meshes. Three types of metric tensor that correspond to uniform and arclength-based and Hessian-based adaptive meshes are considered. The method shows first-order convergence for uniform and arclength-based adaptive meshes, and second-order convergence for Hessian-based adaptive meshes. It is also shown that the method can be used for situations with complex free boundaries, emerging and splitting of free boundaries, and the porous medium equation with variable exponents and absorption. Two-dimensional numerical results are presented.'
author:
- 'Cuong Ngo [^1]'
- 'Weizhang Huang [^2]'
title: |
    A Study on Moving Mesh Finite Element Solution of\
    the Porous Medium Equation 
---

porous medium equation, adaptive moving mesh method, MMPDE method, finite element method, Hessian-based adaptivity, immersed boundary, free boundary

65M60, 65M50, 35Q35

Introduction {#SEC:intro}
============

We consider the numerical solution of 