In this study, we evaluate the performance of feedback control-based time step adaptivity schemes for the nonlocal Cahn-Hilliard equation derived from the Ohta-Kawasaki free energy functional. The temporal adaptivity scheme is recast under the linear feedback control theory equipped with an error estimation that extrapolates the solution obtained from an energy-stable, fully implicit time marching scheme. We test three time step controllers with different properties: a simple Integral controller, a complete Proportional-Integral-Derivative controller, and the PC11 predictive controller. We assess the performance of the adaptive schemes for the nonlocal Cahn-Hilliard equation in terms of the number of time steps required for the complete simulation and the computational effort measured by the required number of nonlinear and linear solver iterations. We also present numerical evidence of mass conservation and free energy decay for simulations with the three different time step controllers. The PC11 predictive controller is the best in all three-dimensional test cases.

中文翻译：

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具有反馈控制时间步长自适应的非局部 Cahn-Hilliard 方程的有限元解

在这项研究中，我们评估了基于反馈控制的时间步长自适应方案的性能，用于从 Ohta-Kawasaki 自由能泛函导出的非局部 Cahn-Hilliard 方程。时间自适应方案是在线性反馈控制理论下重新制定的，该理论配备了误差估计，可以推断从能量稳定、完全隐式时间推进方案获得的解决方案。我们测试了三个具有不同属性的时间步控制器：一个简单的积分控制器、一个完整的比例-积分-微分控制器和 PC11 预测控制器。我们根据完整模拟所需的时间步长数以及所需非线性和线性求解器迭代次数所测量的计算工作量来评估非局部 Cahn-Hilliard 方程的自适应方案的性能。我们还为使用三种不同时间步长控制器的模拟提供了质量守恒和自由能衰减的数值证据。PC11 预测控制器是所有三维测试用例中最好的。