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Newton's method and its hybrid with machine learning for Navier-Stokes Darcy Models discretized by mixed element methods

Jianguo Huang, Hui Peng, Haohao Wu

TL;DR

An Int-Deep algorithm is constructed by combining the previous two methods so as to further improve the computational efficiency and robustness of the Newton iterative method for the steady state Navier-Stokes Darcy model discretized by mixed element methods.

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.

Newton's method and its hybrid with machine learning for Navier-Stokes Darcy Models discretized by mixed element methods

TL;DR

An Int-Deep algorithm is constructed by combining the previous two methods so as to further improve the computational efficiency and robustness of the Newton iterative method for the steady state Navier-Stokes Darcy model discretized by mixed element methods.

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.
Paper Structure (15 sections, 7 theorems, 62 equations, 2 figures, 11 tables, 2 algorithms)

This paper contains 15 sections, 7 theorems, 62 equations, 2 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Navier-Stokes Darcy problem and its discretization
  3. Navier-Stokes Darcy model
  4. Finite element approximation
  5. Newton's Method for the Navier-Stokes Darcy model
  6. The Int-Deep method
  7. Deep learning method
  8. The Int-Deep method for solving the Navier-Stokes Darcy problem
  9. Numerical experiments
  10. A 2D--example with a closed-form solution
  11. The Performance of Algorithm \ref{['alg:DL']}
  12. The performance of Algorithm \ref{['alg:int-deep']}
  13. A 2D-example without closed form solution
  14. A 3D-example with closed form solution
  15. Conclusion

Key Result

Theorem 2.1

\newlabelFEM-well-pose0 Assume that $\bm{f}_f\in [L^2(\Omega_f)]^d$, $f_p\in L^2(\Omega_p)$ and the viscosity coefficient $\nu$ satisfies with Then the finite element scheme $(FEM-1)-(FEM-2)$ has a unique solution satisfying the following estimate

Figures (2)

  • Figure 1: Domain schematic for Naiver-Stokes Darcy coupled flow.
  • Figure 1: (Left) Numerical velocity and pressure under different initial conditions on a triangular mesh with h =$\frac{1}{16}$; (Right) The streamlines of velocity.

Theorems & Definitions (13)

  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Lemma 3.1
  • Proof 1
  • Lemma 3.2
  • Proof 2
  • Theorem 3.3
  • Proof 3
  • ...and 3 more