  In this paper, we introduce a new approach to solving the porous medium
equation using a moving mesh finite element method that leverages the Onsager
variational principle as an approximation tool. Both the continuous and
discrete problems are formulated based on the Onsager principle. The energy
dissipation structure is maintained in the semi-discrete and fully implicit
discrete schemes. We also develop a fully decoupled explicit scheme by which
only a few linear equations are solved sequentially in each time step. The
numerical schemes exhibit an optimal convergence rate when the initial mesh is
appropriately selected to ensure accurate approximation of the initial data.
Furthermore, the method naturally captures the waiting time phenomena without
requiring any manual intervention.
