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In this work, we present the construction of two distinct finite elementapproaches to solve the Porous Medium Equation (PME). In the first approach, wetransform the PME to a log-density variable formulation and construct acontinuous Galerkin method. In the second approach, we introduce additionalpotential and velocity variables to rewrite the PME into a system of equations,for which we construct a mixed finite element method. Both approaches arefirst-order accurate, mass conserving, and proved to be unconditionally energystable for their respective energies. The mixed approach is shown to preservepositivity under a CFL condition, while a much stronger property ofunconditional bound preservation is proved for the log-density approach. Anovel feature of our schemes is that they can handle compactly supportedinitial data without the need for any perturbation techniques. Furthermore, thelog-density method can handle unstructured grids in any number of dimensions,while the mixed method can handle unstructured grids in two dimensions. Wepresent results from several numerical experiments to demonstrate theseproperties.

Two Finite Element Approaches For The Porous Medium Equation That Are Positivity Preserving And Energy Stable