A Study on Moving Mesh Finite Element Solution of the Porous Medium Equation

TL;DR

This paper develops an adaptive moving mesh finite element method for solving the porous medium equation, demonstrating different convergence rates depending on mesh type and applicability to complex boundary scenarios.

Contribution

It introduces a novel implementation of the moving mesh PDE approach for the porous medium equation with variable exponents and absorption, including diverse mesh strategies.

Findings

First order convergence for uniform and arclength meshes

Second order convergence for Hessian-based meshes

Effective handling of complex free boundary problems

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 so-called moving mesh partial differential equation approach and employs its newly developed implementation. Three types of mesh are considered, uniform and arclength-based and Hessian-based adaptive meshes. The method shows a first order convergence for uniform and arclength-based adaptive meshes and a 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.