Analysis of Fully Discrete Finite Element Methods for a System of Differential Equations Modeling Swelling Dynamics of Polymer Gels

The primary goal of this paper is to develop and analyze some fully discrete finite element methods for a displacement-pressure model modeling swelling dynamics of polymer gels under mechanical constraints. In the model, the swelling dynamics is governed by the solvent permeation and the elastic interaction; the permeation is described by a pressure equation for the solvent, and the elastic interaction is described by displacement equations for the solid network of the gel. The elasticity is of long range nature and gives effects for the solvent diffusion. It is the fluid-solid interaction in the gel network drives the system and makes the problem interesting and difficult. By introducing an “elastic pressure” (or “volume change function”) we first present a reformulation of the original model, we then propose a time-stepping scheme which decouples the PDE system at each time step into two sub-problems, one of which is a generalized Stokes problem for the displacement vector field (of the solid network of the gel) and another is a diffusion problem for a “pseudo-pressure” field (of the solvent of the gel). To make such a multiphysical approach feasible, it is vital to find admissible constraints to resolve the uniqueness issue for the generalized Stokes problem and to construct a “good” boundary condition for the diffusion equation so that it also becomes uniquely solvable. The key to the first difficulty is to discover certain conservation laws (or conserved quantities) for the PDE solution of the original model, and the solution to the second difficulty is to use the generalized Stokes problem to generate a boundary condition for the diffusion problem. This then lays down the theoretical foundation for one to utilize any convergent Stokes solver (and its code) together with any convergent diffusion equation solver (and its code) to solve the polymer gel model. In the paper, the Taylor-Hood mixed finite element method combined with the continuous linear finite element method are chosen as an example to present the ideas and to demonstrate the viability of the proposed multiphysical approach. It is proved that, under a mesh constraint, both the proposed semi-discrete (in space) and fully discrete methods enjoy some discrete energy laws which mimic the differential energy law satisfied by the PDE solution. Optimal order error estimates in various norms are established for the numerical solutions of both the semi-discrete and fully discrete methods. Numerical experiments are also presented to show the efficiency of the proposed approach and methods.