  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.
