A Moving Mesh Method for Porous Medium Equation by the Onsager Variational Principle

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.