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

TL;DR

This paper develops and analyzes fully discrete finite element methods for a displacement-pressure model of polymer gel swelling, introducing an elastic pressure reformulation and a decoupled time-stepping scheme that ensures stability and accuracy.

Contribution

It proposes a novel multiphysical decoupled scheme for polymer gel modeling, allowing flexible use of existing solvers and establishing energy laws and error estimates.

Findings

The scheme is unconditionally stable under mesh constraints.

Optimal error estimates are proved for the numerical solutions.

Numerical experiments confirm the efficiency and accuracy of the methods.

Abstract

The 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. By introducing an "elastic pressure" we first present a reformulation of the original model, and 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 and another is a diffusion problem for a "pseudo-pressure" field. To make such a multiphysical approach feasible, it is vital to discover admissible constraints to resolve the uniqueness issue for both sub-problems. The main advantage of the proposed approach is that it allows one to utilize any convergent Stokes solver together with any convergent diffusion equation solver 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 used 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.