Newton’s Method and Its Hybrid with Machine Learning for Navier-Stokes Darcy Models Discretized by Mixed Element Methods (Communications in Computational Physics, 2025)
This paper focuses on discussing Newton’s method and its hybrid with machine learning for the steady state Navier-Stokes Darcy model discretized by mixed element methods. First, a Newton iterative method is introduced for solving the relativediscretized problem. It is proved technically that this method converges quadraticallywith the convergence rate independent of the mixed element mesh size, under certainstandard conditions. Later on, a deep learning algorithm is proposed for solving thisnonlinear coupled problem. Following the ideas of an earlier work by Huang, Wangand Yang (2020), an Int-Deep algorithm is constructed by combining the previous twomethods so as to further improve the computational efficiency and robustness. A series of numerical examples are reported to show the numerical performance of theproposed methods.