Adaptive Finite Element Solution of the Porous Medium Equation in Pressure Formulation

TL;DR

This paper develops an adaptive finite element method using pressure formulation for the porous medium equation, achieving high accuracy in free boundary tracking and demonstrating advantages over traditional methods in certain scenarios.

Contribution

It introduces the pressure formulation into numerical solutions of the porous medium equation with adaptive mesh refinement, improving free boundary accuracy.

Findings

Second-order spatial accuracy in pressure variable

Almost second-order convergence in free boundary location

Advantages in large exponent cases or precise boundary tracking

Abstract

A lack of regularity in the solution of the porous medium equation poses a serious challenge in its theoretical and numerical studies. A common strategy in theoretical studies is to utilize the pressure formulation of the equation where a new variable called the mathematical pressure is introduced. It is known that the new variable has much better regularity than the original one and Darcy's law for the movement of the free boundary can be expressed naturally in this new variable. The pressure formulation has not been used in numerical studies. The goal of this work is to study its use in the adaptive finite element solution of the porous medium equation. The MMPDE moving mesh strategy is employed for adaptive mesh movement while linear finite elements are used for spatial discretization. The free boundary is traced explicitly by integrating Darcy's law with the Euler scheme. Numerical results are presented for three two-dimensional examples. The method is shown to be second-order in space and first-order in time in the pressure variable. Moreover, the convergence order of the error in the location of the free boundary is almost second-order in the maximum norm. However, numerical results also show that the convergence order of the method in the original variable stays between first-order and second-order in the $L^1$ norm or between 0.5th-order and first-order in the $L^2$ norm. Nevertheless, the current method can offer some advantages over numerical methods based on the original formulation for situations with large exponents or when a more accurate location of the free boundary is desired.