聚合物凝胶膨胀动力学微分方程组的全离散有限元方法分析
Analysis of Fully Discrete Finite Element Methods for a System of Differential Equations Modeling Swelling Dynamics of Polymer Gels
AI总结：
针对机械约束下聚合物凝胶膨胀动力学模型，提出一种通过引入弹性压力解耦为广义Stokes问题和扩散问题的多物理场时间步进方案，并采用Taylor-Hood混合有限元与连续线性有限元结合进行离散，证明了半离散和全离散方法的离散能量守恒律及最优阶误差估计。
摘要：
本文的目标是发展和分析一些全离散有限元方法，用于模拟机械约束下聚合物凝胶膨胀动力学的位移-压力模型。在该模型中，膨胀动力学由溶剂渗透和弹性相互作用控制；渗透由溶剂的压力方程描述，弹性相互作用由凝胶固体网络的位移方程描述。通过引入“弹性压力”，我们首先对原始模型进行重新表述，然后提出一个时间步进方案，该方案在每个时间步将PDE系统解耦为两个子问题：一个是位移向量场的广义Stokes问题，另一个是“伪压力”场的扩散问题。为了使这种多物理场方法可行，发现可接受的约束条件以解决两个子问题的唯一性问题至关重要。所提出方法的主要优点是允许利用任何收敛的Stokes求解器和任何收敛的扩散方程求解器来求解聚合物凝胶模型。本文以Taylor-Hood混合有限元方法与连续线性有限元方法结合为例，展示思路并证明所提出的多物理场方法的可行性。证明了在网格约束下，所提出的半离散（空间）和全离散方法都满足一些离散能量守恒律，这些定律模仿了PDE解满足的微分能量守恒律。为半离散和全离散方法的数值解建立了各种范数下的最优阶误差估计。还给出了数值实验，以展示所提出方法和方法的效率。
英文摘要：
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.
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