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Continuous-data-assimilation-enabled fast and robust convergence of an Uzawa-based solver for Navier-Stokes equations with large Reynolds number

Victoria Luongo Fisher, Jessica C. Franklin, Leo G. Rebholz

Abstract

This paper shows how continuous data assimilation (CDA) can be used to provably enable and accelerate convergence of a (efficient at each iteration due to a physics-splitting, but generally slowly converging and not robust) nonlinear solver for incompressible Navier-Stokes equations (NSE). Herein we develop, analyze and test an Uzawa-based nonlinear solver for incompressible NSE that incorporates partial solution data into the iteration through continuous data assimilation (CDA-Uzawa). We rigorously prove that i) CDA-Uzawa will accelerate a converging Uzawa iteration, and more partial solution data yields more acceleration, and ii) with enough partial solution data CDA-Uzawa will converge for arbitrarily large Reynolds numbers, even if multiple NSE solutions exist. In the case of noisy data, we prove that the convergence results hold down to the size of the noise, and we propose a strategy to pass to Newton once CDA-Uzawa convergence reaches its lower limit. Results of several numerical tests illustrate the theory and show CDA-Uzawa is a very effective and efficient solver. While this paper focuses a particular splitting-based solver for the NSE, the key ideas are quite general and extendable to a wide class of nonlinear solvers.

Continuous-data-assimilation-enabled fast and robust convergence of an Uzawa-based solver for Navier-Stokes equations with large Reynolds number

Abstract

This paper shows how continuous data assimilation (CDA) can be used to provably enable and accelerate convergence of a (efficient at each iteration due to a physics-splitting, but generally slowly converging and not robust) nonlinear solver for incompressible Navier-Stokes equations (NSE). Herein we develop, analyze and test an Uzawa-based nonlinear solver for incompressible NSE that incorporates partial solution data into the iteration through continuous data assimilation (CDA-Uzawa). We rigorously prove that i) CDA-Uzawa will accelerate a converging Uzawa iteration, and more partial solution data yields more acceleration, and ii) with enough partial solution data CDA-Uzawa will converge for arbitrarily large Reynolds numbers, even if multiple NSE solutions exist. In the case of noisy data, we prove that the convergence results hold down to the size of the noise, and we propose a strategy to pass to Newton once CDA-Uzawa convergence reaches its lower limit. Results of several numerical tests illustrate the theory and show CDA-Uzawa is a very effective and efficient solver. While this paper focuses a particular splitting-based solver for the NSE, the key ideas are quite general and extendable to a wide class of nonlinear solvers.
Paper Structure (22 sections, 2 theorems, 60 equations, 14 figures)

This paper contains 22 sections, 2 theorems, 60 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. NSE Preliminaries
  4. Finite Element Preliminaries
  5. Uzawa
  6. CDA Preliminaries
  7. CDA-Uzawa
  8. Analysis of CDA-Uzawa
  9. The case of noisy data
  10. Numerical Experiments
  11. Test problem descriptions
  12. 2D driven cavity
  13. 3D driven cavity
  14. Flow in a stenotic artery
  15. 2D channel flow past a block
  16. ...and 7 more sections

Key Result

Theorem 3.1

Let $(u,p)\in (X,Q)$ solve the NSE eq4-eq5, and suppose that $I_H(u)$ is known. Let $(u_{k+1},p_{k+1})\in (X,Q)$ be the $k+1^{st}$ iterate of the CDA-Uzawa algorithm CDAU1-CDAU2. Suppose $H$ is chosen small enough and $\gamma$ large enough so that $\kappa<1$, and $\mu \ge \frac{\nu}{2C_I^2H^2}$. The

Figures (14)

  • Figure 1: The plots above show streamlines of the solution of the 2D driven cavity problem solution at varying $Re$ taken from our finest mesh simulations.
  • Figure 2: The plots above show $Re$=1000 (top) and $Re$=2000 (bottom) 3D driven cavity velocity solutions displayed as midspliceplanes of velocity.
  • Figure 3: Shown above is the artery mesh (before the barycenter refinement is applied) restricted to the surface.
  • Figure 4: Shown above are contour slices of the velocity magnitude for the $\nu=\frac{1}{500}$ solution found using Anderson accelerated Picard followed by Newton iterations until $H^1$ convergence of successive iterates to $10^{-10}$.
  • Figure 5: Shown above is a sample mesh for 2D channel flow past a block.
  • ...and 9 more figures

Theorems & Definitions (6)

  • Remark 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Remark 3.1
  • proof