Mixed Finite Element Formulation and Error Estimates Based on Proper Orthogonal Decomposition for the Nonstationary Navier-Stokes Equations

In this paper, proper orthogonal decomposition (POD) is used for model reduction of mixed finite element (MFE) for the nonstationary Navier-Stokes equations and error estimates between a reference solution and the POD solution of reduced MFE formulation are derived. The basic idea of this reduction technique is that ensembles of data are first compiled from transient solutions computed equation system derived with the usual MFE method for the nonstationary Navier-Stokes equations or from physics system trajectories by drawing samples of experiments and interpolation (or data assimilation), and then the basis functions of the usual MFE method are substituted with the POD basis functions reconstructed by the elements of the ensemble to derive the POD-reduced MFE formulation for the nonstationary Navier-Stokes equations. It is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between POD solution of reduced MFE formulation and the reference solution is consistent with theoretical results. Moreover, it is also shown that this result validates the feasibility and efficiency of the POD method.

• Publication year

  2008

• Venue

  SIAM Journal on Numerical Analysis

• Publication date

  2008-10-31

• Fields of study

  Mathematics, Physics, Computer Science, Engineering

• Identifiers
  DOI 10.1137/070689498
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Turbulence

  2017cited by this paper
• NUMERICAL SIMULATION

  2016cited by this paper
• Mixed and Hybrid Finite Element Methods

  2011cited by this paper
• An Equation-Free Reduced-Order Modeling Approach to Tropical Pacific Simulation

  2009cited by this paper
• An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model

  2007cited by this paper
• A reduced‐order approach to four‐dimensional variational data assimilation using proper orthogonal decomposition

  2007cited by this paper
• Finite difference scheme based on proper orthogonal decomposition for the nonstationary Navier-Stokes equations

  2007cited by this paper
• Reduced-Order Modeling of the Upper Tropical Pacific Ocean Model using Proper Orthogonal Decomposition

  2006cited by this paper
• Hodge Duality and Cosmological Constant

  2004cited by this paper
• Systematic Strategies for Stochastic Mode Reduction in Climate

  2003cited by this paper
• Proper Orthogonal Decomposition for Flow Calculations and Optimal Control in a Horizontal CVD Reactor

  2002cited by this paper
• Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics

  2002influential reference
• Proper Orthogonal Decomposition for Time-Dependent Lid-Driven Cavity Flows

  2002cited by this paper
• The finite element method for elliptic problems

  2002cited by this paper
• Principal Component Analysis

  2002cited by this paper
• SYNTHETIC JETS

  2001cited by this paper
• Galerkin proper orthogonal decomposition methods for parabolic problems

  2001cited by this paper
• Synthetic jets, their reduced order modeling and applications to flow control

  1999cited by this paper
• Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition

  1999cited by this paper
• Proper orthogonal decomposition and low-dimensional models for driven cavity flows

  1998cited by this paper
• Baroclinic empirical orthogonal functions as basis functions in an atmospheric model

  1997cited by this paper
• A Self-Contained, Automated Methodology for Optimal Flow Control

  1997cited by this paper
• A SELF-CONTAINED, AUTOMATED METHODOLOGY FOR OPTIMAL FLOW CONTROL VALIDATED FOR TRANSITION DELAY

  1995cited by this paper
• Low-dimensional description of free-shear-flow coherent structures and their dynamical behaviour

  1994cited by this paper
• The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows

  1993cited by this paper
• Characteristic-eddy decomposition of turbulence in a channel

  1989cited by this paper
• The dynamics of coherent structures in the wall region of a turbulent boundary layer

  1988cited by this paper
• Turbulence and the dynamics of coherent structures. I. Coherent structures

  1987cited by this paper
• Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms

  1986influential reference
• Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization

  1982cited by this paper
• Coherent Structures in Turbulence

  1981influential reference
• On the finite element approximation of the nonstationary Navier-Stokes problem

  1980influential reference

• Nonconforming finite element method and reduced order algorithm for poroelasticity problem

  2026cites this paper
• A POD-enhanced simplified weak Galerkin method for solute transport problems

  2026cites this paper
• Two-grid reduced-dimension method and applications for the nonlinear single phase flow model coupled with ground stress in porous median

  2026cites this paper
• Reduced-Dimension Study of Two-Grid Mixed Finite Element Crank-Nicolson Algorithm for the Fourth-Order Nonlinear Rosenau Equation

  2025cites this paper
• Efficient identification of geometric inverse sources of parabolic problems by model order reduction

  2025cites this paper
• A Priori Analysis of a Tensor ROM for Parameter Dependent Parabolic Problems

  2025cites this paper
• A Priori Error Bounds for the Approximate Deconvolution Leray Reduced Order Model

  2024cites this paper
• Research on fine analysis and accelerated prediction technology for thermal stratification in the upper plenum of the “Monju” reactor

  2024cites this paper
• A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term

  2024cites this paper
• A Mass-Conservative Reduced-Order Algorithm in Solving Optimal Control of Convection-Diffusion Equation

  2024cites this paper
• A POD based extrapolation DG time stepping space-time FE method for parabolic problems

  2024cites this paper
• A reduced-order Schwarz domain decomposition method based on POD for the convection-diffusion equation

  2024cites this paper
• A reduced-order least squares-support vector regression and isogeometric collocation method to simulate Cahn-Hilliard-Navier-Stokes equation

  2024cites this paper
• A stabilizer-free weak Galerkin finite element method for an optimal control problem of a time fractional diffusion equation

  2024cites this paper
• Two-grid reduced-order method based on POD for a nonlinear poroelasticity model

  2024cites this paper
• Reduced‐order extrapolation technique for coefficient vectors of numerical solutions about fluid mechanics equation

  2023cites this paper
• A reduced-order Jacobi spectral collocation method for solving the space-fractional FitzHugh–Nagumo models with application in myocardium

  2023cites this paper
• A Pressure-Stabilized Continuous Data Assimilation Reduced Order Model for Incompressible Navier–Stokes Equations

  2023cites this paper
• Investigation of phase-field models of tumor growth based on a reduced-order meshless Galerkin method

  2023cites this paper
• A priori analysis of a tensor ROM for parameter dependent parabolic problems

  2023cites this paper
• A reduced-order model based on C-R mixed finite element and POD technique for coupled Stokes-Darcy system with solute transport

  2023cites this paper
• Hybridizable discontinuous Galerkin reduced order model for the variable coefficient advection equation

  2023cites this paper
• Reduced-order finite element approximation based on POD for the parabolic optimal control problem

  2023cites this paper
• Reduced order model for simulation of air pollution model and application in 2D urban street canyons via the meshfree gradient smoothing method

  2023cites this paper
• A reduced-order immersed interface method based on POD basis for parabolic interface problem

  2023cites this paper
• A reduced-order model based on cubic B-spline basis function and SSP Runge–Kutta procedure to investigate option pricing under jump-diffusion models

  2023cites this paper
• Parameter Estimation for a Kinetic Model of a Cellular System Using Model Order Reduction Method

  2023cites this paper
• Proper Orthogonal Decomposition-Lattice Boltzmann Method: Simulating the Air Pollutant Problem in Street Canyon Areas

  2022cites this paper
• The reduced-order method of continuous space-time finite element scheme for the non-stationary incompressible flows

  2022cites this paper
• An Efficient Chorin–Temam Projection Proper Orthogonal Decomposition Based Reduced-Order Model for Nonstationary Stokes Equations

  2022cites this paper
• A reduced-order characteristic finite element method based on POD for optimal control problem governed by convection–diffusion equation

  2022cites this paper
• Numerical simulation for a incompressible miscible displacement problem using a reduced-order finite element method based on POD technique

  2021cites this paper
• The proper orthogonal decomposition modal spectral element method for two-dimensional viscoelastic equation

  2021cites this paper
• A new approach to proper orthogonal decomposition with difference quotients

  2021cites this paper
• A fast finite difference/RBF meshless approach for time fractional convection-diffusion equation with non-smooth solution

  2021cites this paper
• A reduced-order Weak Galerkin finite element algorithm based on POD technique for parabolic problem on polytopal mesh

  2021cites this paper
• Data-Driven Reduced-Order Modeling of Convective Heat Transfer in Porous Media

  2021cites this paper
• Reduced order modeling of time-dependent incompressible Navier–Stokes equation with variable density based on a local radial basis functions-finite difference (LRBF-FD) technique and the POD/DEIM method

  2020cites this paper
• A reduced-order extrapolated Crank–Nicolson finite spectral element method for the 2D non-stationary Navier-Stokes equations about vorticity-stream functions

  2020cites this paper
• A POD-based reduced-order Crank-Nicolson/fourth-order alternating direction implicit (ADI) finite difference scheme for solving the two-dimensional distributed-order Riesz space-fractional diffusion equation

  2020cites this paper
• Simulation flows with multiple phases and components via the radial basis functions-finite difference (RBF-FD) procedure: Shan-Chen model

  2020cites this paper
• On Optimal Pointwise in Time Error Bounds and Difference Quotients for the Proper Orthogonal Decomposition

  2020cites this paper
• A proper orthogonal decomposition variational multiscale meshless interpolating element‐free Galerkin method for incompressible magnetohydrodynamics flow

  2020cites this paper
• A highly efficient reduced-order extrapolated finite difference algorithm for time-space tempered fractional diffusion-wave equation

  2020cites this paper
• Foreword and Introduction

  2019cites this paper
• A reduced-order extrapolating collocation spectral method based on POD for the 2D Sobolev equations

  2019cites this paper
• A reduced-order extrapolated Crank–Nicolson finite spectral element method based on POD for the 2D non-stationary Boussinesq equations

  2019cites this paper
• Proper orthogonal decomposition with SUPG-stabilized isogeometric analysis for reduced order modelling of unsteady convection-dominated convection-diffusion-reaction problems

  2019cites this paper
• A reduced‐order extrapolated natural boundary element method based on POD for the 2D hyperbolic equation in unbounded domain

  2019cites this paper
• A reduced-order extrapolated natural boundary element method based on POD for the parabolic equation in the 2D unbounded domain

  2019cites this paper
• An optimized Crank–Nicolson finite difference extrapolating model for the fractional-order parabolic-type sine-Gordon equation

  2019cites this paper
• A Reduced High-order Compact Finite Difference Scheme Based on Proper Orthogonal Decomposition for the Generalized Kuramoto-Sivashinsky Equation

  2019cites this paper
• A reduced‐order extrapolated Crank–Nicolson collocation spectral method based on proper orthogonal decomposition for the two‐dimensional viscoelastic wave equations

  2019cites this paper
• Numerical analysis of a projection-based stabilized POD-ROM for incompressible flows

  2019cites this paper
• A reduced fourth-order compact difference scheme based on a proper orthogonal decomposition technique for parabolic equations

  2019cites this paper
• Galerkin proper orthogonal decomposition-reduced order method (POD-ROM) for solving generalized Swift-Hohenberg equation

  2019cites this paper
• Reduced-Order Extrapolation Finite Volume Element Methods Based on Proper Orthogonal Decomposition

  2019cites this paper
• Bibliography

  2019cites this paper
• Crank–Nicolson Finite Element Scheme and Modified Reduced-Order Scheme for Fractional Sobolev Equation

  2018cites this paper
• Model reduction of dynamical systems by proper orthogonal decomposition: Error bounds and comparison of methods using snapshots from the solution and the time derivatives

  2018cites this paper
• An optimized finite element extrapolating method for 2D viscoelastic wave equation

  2017cites this paper
• Large Eddy Simulation Reduced Order Models

  2017cites this paper
• Energy balance and mass conservation in reduced order models of fluid flows

  2017cites this paper
• Isogeometric analysis and proper orthogonal decomposition for the acoustic wave equation

  2017cites this paper
• HDG-POD reduced order model of the heat equation

  2017cites this paper
• A stabilized MFE reduced-order extrapolation model based on POD for the 2D unsteady conduction-convection problem

  2017cites this paper
• Streamline derivative projection-based POD-ROM for convection-dominated flows. Part I : Numerical Analysis

  2017cites this paper
• Quantile Regression Analysis on Short-Term Measured Data of Gravity-Arch Dam Based on POD

  2017cites this paper
• Numerical analysis of the Leray reduced order model

  2017cites this paper
• An optimized SPDMFE extrapolation approach based on the POD technique for 2D viscoelastic wave equation

  2017cites this paper
• Reduced-order proper orthogonal decomposition extrapolating finite volume element format for two-dimensional hyperbolic equations

  2017cites this paper
• The use of proper orthogonal decomposition (POD) meshless RBF-FD technique to simulate the shallow water equations

  2017cites this paper
• Optimized finite difference iterative scheme based on POD technique for 2D viscoelastic wave equation

  2017cites this paper
• Model reduction and inverse problems and data assimilation with geophysical applications. A special issue in honor of I. Michael Navon's 75th birthday

  2016cites this paper
• Numerical and Analytical Aspects of POD-Based Reduced-Order Modeling in Computational Fluid Dynamics

  2016cites this paper
• Proper orthogonal decomposition variational multiscale element free Galerkin (POD-VMEFG) meshless method for solving incompressible Navier–Stokes equation

  2016cites this paper
• A reduced-order extrapolation central difference scheme based on POD for two-dimensional fourth-order hyperbolic equations

  2016cites this paper
• A POD reduced-order finite difference time-domain extrapolating scheme for the 2D Maxwell equations in a lossy medium

  2016cites this paper
• Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD-Galerkin method and a vascular shape parametrization

  2016cites this paper
• Isogeometric analysis and proper orthogonal decomposition for parabolic problems

  2016cites this paper
• Reduced-order extrapolation spectral-finite difference scheme based on POD method and error estimation for three-dimensional parabolic equation

  2015cites this paper
• A POD-Based Reduced-Order Stabilized Crank–Nicolson MFE Formulation for the Non-Stationary Parabolized Navier–Stokes Equations

  2015cites this paper
• A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations

  2015cites this paper
• A POD-based reduced-order TSCFE extrapolation iterative format for two-dimensional heat equations

  2015cites this paper
• Application of low-dimensional finite element method to fractional diffusion equation

  2014cites this paper
• A numerical investigation of velocity-pressure reduced order models for incompressible flows

  2014cites this paper
• A reduced-order extrapolation algorithm based on CNLSMFE formulation and POD technique for two-dimensional Sobolev equations

  2014cites this paper
• A reduced-order extrapolation algorithm based on SFVE method and POD technique for non-stationary Stokes equations

  2014cites this paper
• New POD Error Expressions, Error Bounds, and Asymptotic Results for Reduced Order Models of Parabolic PDEs

  2014cites this paper
• A reduced-order finite volume element formulation based on POD method and numerical simulation for two-dimensional solute transport problems

  2013cites this paper
• A reduced-order Crank-Nicolson finite volume element formulation based on POD method for parabolic equations

  2013cites this paper
• Nonlinear model reduction based on the finite element method with interpolated coefficients: Semilinear parabolic equations

  2013cites this paper
• Burgers方程基于POD方法的降维CN有限元外推算法 A Reduced-Order CN Finite Element Extrapolating Algorithm Based on POD for Burgers Equation

  2013cites this paper
• Are the Snapshot Difference Quotients Needed in the Proper Orthogonal Decomposition?

  2013cites this paper
• Variational multiscale proper orthogonal decomposition: Convection-dominated convection-diffusion-reaction equations

  2013cites this paper
• Computational Parabolic Inverse Problems

  2012cites this paper
• Variational multiscale proper orthogonal decomposition: Navier‐stokes equations

  2012cites this paper
• An optimizing implicit difference scheme based on proper orthogonal decomposition for the two‐dimensional unsaturated soil water flow equation

  2012cites this paper
• A reduced FVE formulation based on POD method and error analysis for two-dimensional viscoelastic problem☆

  2012cites this paper
• A reduced-order LSMFE formulation based on POD method and implementation of algorithm for parabolic equations

  2012cites this paper