Fully discrete stabilized multiphysics finite element method for the polymer gel model

In this paper, we propose a fully discrete stabilized multiphysics finite element method by the pressure projection method to solve the polymer gel model. To reveal the multi-physical process of the polymer gel model, we reformulate the polymer gel model by introducing a new variable to get a new multi-field problem, then we propose the fully discrete stabilized multiphysics finite element method using the simplest equal order element of P1−P1−P1 without the discrete inf-sup condition to solve the new reformulated system, and we prove that the new method not only has no ”locking” phenomenon, but also preserves some energy conservation laws. Also, we prove that the stabilized multiphysics finite element method has optimal convergence order. Finally, we give the numerical example to verify the theoretical results.