NEWTON’S METHOD AND ITS HYBRID WITH MACHINE LEARNING FOR NAVIER-STOKES DARCY MODELS DISCRETIZED BY MIXED ELEMENT METHODS

Aug 6, 2026

Steady-state Navier-Stokes Darcy model discretized by mixed element methods is addressed through a hybrid numerical strategy combining Newton iterations with machine learning. Newton’s method is formulated for the resulting nonlinear coupled system, with a rigorous proof of quadratic convergence whose rate is independent of the finite element mesh size under standard assumptions. To enhance efficiency and robustness, a deep learning-based Int-Deep algorithm couples the learned nonlinear solver with the Newton framework. Numerical experiments validate the performance of the proposed methods.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages23

Reading Time58 min

AccessFree to download

Introduction

Background and coupled strategies

Iterative methods and convergence challenges

Newton’s method for the mixed-element discretization

Convergence properties and mesh independence

Deep learning hybridization

Int-Deep algorithm

Numerical experiments

Computational performance and robustness

arXiv :2401 . 10557v2 [math .NA] 6 Mar 2024 NEWTON’S METHOD AND ITS HYBRID WITH MACHINE LEARNING FOR NAVIER-STOKES DARCY MODELS DISCRETIZED BY MIXED ELEMENT METHODS ∗ JIANGUO HUANG†, HUI PENG†, AND HAOHAO WU† Abstract. 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 relative discretized problem. It is proved technically that this method converges quadratically with the convergence rate independent of the finite element mesh size, under certain standard conditions. Later on, a deep learning algorithm is proposed for solving this nonlinear coupled problem. Following the ideas of an earlier work by Huang, Wang and Yang (2020), an Int-Deep algorithm is constructed by combining the previous two methods so as to further improve the computational efficiency and robustness. A series of numerical examples are reported to show the numerical performance of the proposed methods. Key words. Navier Stokes-Darcy model · Deep learning · Newton iterative method · Int-Deep method · Convergence analysis AMS subject classifications. 65N12, 65N15, 65N22, 65N30 1. Introduction. The Navier-Stokes Darcy model is frequently encountered in various industrial engineering scenarios, for instance, in groundwater [10, 14, 26], flow in porous media [3, 2], industrial filtrations [17] and so on. Due to its importance, many numerical methods have been developed and analyzed for the coupled nonlinear model, which can be roughly divided into two categories. The first one focuses on numerically solving the coupled system directly. Historically, Girault and Rivi´ere [16] propose and analyze a discontinuous Galerkin method for the Navier-Stokes Darcy model. Following [16], Chidyagwai and Rivi´ere [12] develop a hybrid coupled method, where the continuous finite elements are employed in the free flow region and the discontinuous Galerkin finite elements in the porous medium region, respectively. Wu and Mei [31] discretize the model with a non-conforming finite volume element method. Cesmelioglu and Rhebergen present and analyze in [11] a strongly conservative hybridizable discontinuous Galerkin scheme. The other strategy is to decouple the model first and then numerically solve two subproblems separately. In [9], Cai, Mu and Xu propose a decoupled and linearized two-grid algorithm. The two-grid decoupled method is further studied in [34, 24] . In addition, Cai, Huang and Mu also develop a multi-grid algorithm in [8] . He et al. [19] design a domain decomposition method for the Navier-Stokes Darcy model with the BJ interface condition. In order to further improve computational efficiency, Du et al. develop in [30, 15] a series of parallel algorithms based on decoupled model. We mention that all the previous methods are used to solve steady-state Navier-Stokes Darcy models. In fact, there are deep and through works on numerical solution for unsteady state models, and we skip the details due to the limit of space. Both the direct and decoupling methods require solving nonlinear system of equa- ∗ Submitted to the editors DATE. Funding: The work of the first author was supported by the National Natural Science Foundation of China (Grant No. 12071289), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA25010402) . The work of the second author was supported by the National Natural Science Foundation of China (Grant No. 12301519) †School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China ([jghuang@sjtu.edu.cn](jghuang@sjtu.edu.cn), [penghui23@sjtu.edu.cn](penghui23@sjtu.edu.cn), [wu1150132305@sjtu.edu.cn](wu1150132305@sjtu.edu.cn)). 1 This manuscript is for review purposes only. tions globally or locally through iterative methods, e.g. Newton’s method. However, there are few works on investig

FAQs

What problem does the paper study?

The paper studies steady-state Navier-Stokes Darcy models discretized by mixed element methods, focusing on solving the resulting nonlinear coupled system efficiently and robustly.

How is Newton’s method used in this work?

Newton’s iterative method is introduced for the relative discretized problem, and it is proved to converge quadratically under standard conditions.

What is the role of machine learning in the proposed approach?

A deep learning algorithm is proposed to handle the nonlinear coupled problem, and the Int-Deep method combines the Newton-based solver with the learned component to improve computational efficiency and robustness.

Aug 6, 2026

Upload by Lucas Martin