A moving mesh method for porous medium equation by the Onsager variational principle

Abstract

In this paper, we present a novel moving mesh finite element method for solving the porous medium equation, using the Onsager variational principle as an approximation framework. We first demonstrate that a mixed formulation of the continuous problem can be derived by applying the Onsager principle. Subsequently, we develop several numerical schemes by approximating the problem within a nonlinear finite element space with free knots (movable nodes), following the same variational approach. We rigorously prove that the energy dissipation structure is preserved in both semi-discrete and fully implicit discrete schemes. Additionally, we propose a fully decoupled explicit scheme, which requires only the sequential solution of a few linear equations per time step. Other variants of the method can also be derived analogously to preserve mass conservation or to enhance stability. The numerical schemes achieve optimal convergence rates when the initial mesh is carefully chosen to ensure good approximation of the initial data. Through extensive numerical experiments, we evaluated and compared the efficiency and stability of the proposed schemes with existing approaches. For cases involving uniform initial meshes, all schemes exhibit good stability, with the fully decoupled scheme demonstrating superior computational efficiency. In contrast, when addressing singular problems on nonuniform meshes, the stabilized explicit scheme strikes a good balance between stability and computational efficiency. In addition, the method inherently captures the waiting time phenomenon without requiring user intervention, further illustrating its robustness.