A discontinuous Galerkin finite element method for swelling model of polymer gels

An attractive feature of discontinuous Galerkin (DG) finite element schemes is that this concept offers a unified and versatile discretization platform for various types of partial differential equations. The locality of the trial functions not only supports local mesh refinements but also offers a framework for comfortably varying the order of the discretization. In this paper, we propose and analyze a mixed-DG finite element method for a displacement-pressure model which describes swelling dynamics of polymer gels under mechanical constraints. By introducing a flux variable we first present a reformulation of the governing equations of polymer gels. We then approximate the pressure and flux variables by a mixed finite element space and the displacement by DG finite element method. Existence and uniqueness are proved and error estimates are derived for mixed-DG finite element scheme. Finally, numerical experiments are presented to show the performance of the mixed-DG approximation of polymer gels.