We consider systematic numerical approximation of a viscoelastic phase separation model that describes the demixing of a polymer solvent mixture. An unconditionally stable discretisation method is proposed based on a finite element approximation in space and a variational time discretization strategy. The proposed method preserves the energy-dissipation structure of the underlying system exactly and allows to establish a fully discrete nonlinear stability estimate in natural norms based on the concept of relative energy. These estimates are used to derive order optimal error estimates for the method under minimal smoothness assumptions on the problem data, despite the presence of various strong nonlinearities in the equations. The theoretical results and main properties of the method are illustrated by numerical simulations which also demonstrate the capability to reproduce the relevant physical effects observed in experiments.

翻译：我们研究一种描述聚合物溶剂混合物相分离的粘弹性相分离模型的系统数值逼近方法。基于空间有限元逼近和变分时间离散化策略，提出了一种无条件稳定的离散化方法。该方法精确保持了原系统的能量耗散结构，并允许基于相对能量概念在自然范数下建立全离散非线性稳定性估计。尽管方程中存在多种强非线性项，这些估计仍被用于在问题数据满足最小光滑性假设的条件下，推导出该方法阶数最优的误差估计。数值模拟验证了理论结果与方法的主要性质，同时证明了该方法能够复现实验中观测到的相关物理效应。