A stabilized MFE reduced-order extrapolation model based on POD for the 2D unsteady conduction-convection problem

In this study, we devote ourselves to establishing a stabilized mixed finite element (MFE) reduced-order extrapolation (SMFEROE) model holding seldom unknowns for the two-dimensional (2D) unsteady conduction-convection problem via the proper orthogonal decomposition (POD) technique, analyzing the existence and uniqueness and the stability as well as the convergence of the SMFEROE solutions and validating the correctness and dependability of the SMFEROE model by means of numerical simulations.

[1]  Xie Zheng-hui,et al.  A reduced MFE formulation based on POD for the non-stationary conduction-convection problems , 2011 .

[2]  Zhendong Luo,et al.  An optimized SPDMFE extrapolation approach based on the POD technique for 2D viscoelastic wave equation , 2017 .

[3]  Über eine Klasse nichtlinearer Differentialgleichungen im Hilbertraum , 1973 .

[4]  Zhendong Luo A POD-Based Reduced-Order Stabilized Crank–Nicolson MFE Formulation for the Non-Stationary Parabolized Navier–Stokes Equations , 2015 .

[5]  Azzeddine Soulaïmani,et al.  A POD-based reduced-order model for free surface shallow water flows over real bathymetries for Monte-Carlo-type applications , 2012 .

[6]  Zhendong Luo,et al.  A reduced finite element formulation based on POD method for two-dimensional solute transport problems , 2012 .

[8]  Ionel Michael Navon,et al.  Proper orthogonal decomposition approach and error estimation of mixed finite element methods for the tropical Pacific Ocean reduced gravity model , 2007 .

[9]  Cheng Wang,et al.  Analysis of a fourth order finite difference method for the incompressible Boussinesq equations , 2004, Numerische Mathematik.

[10]  Zhendong Luo,et al.  A reduced-order extrapolation algorithm based on CNLSMFE formulation and POD technique for two-dimensional Sobolev equations , 2014 .

[11]  Ping Sun,et al.  Some reduced finite difference schemes based on a proper orthogonal decomposition technique for parabolic equations , 2010 .

[12]  Jing Chen,et al.  An optimizing reduced PLSMFE formulation for non‐stationary conduction–convection problems , 2009 .

[13]  G. Burton Sobolev Spaces , 2013 .

[14]  Zhendong Luo,et al.  A fully discrete stabilized mixed finite volume element formulation for the non-stationary conduction–convection problem , 2013 .

[15]  Adrian Sandu,et al.  Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations , 2014, International Journal for Numerical Methods in Fluids.

[16]  Yinnian He,et al.  A stabilized finite element method based on two local Gauss integrations for the Stokes equations , 2008 .

[17]  Yogendra Joshi,et al.  Error estimation in POD-based dynamic reduced-order thermal modeling of data centers , 2013 .

[18]  Ionel M. Navon,et al.  MIXED FINITE ELEMENT FORMULATION AND ERROR ESTIMATES BASED ON PROPER ORTHOGONAL DECOMPOSITION FOR THE NON-STATIONARY NAVIER–STOKES EQUATIONS* , 2008 .

[19]  Masayuki Yano,et al.  A Space-Time Petrov-Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations , 2014, SIAM J. Sci. Comput..

[20]  Jing Chen,et al.  Mixed Finite Element Formulation and Error Estimates Based on Proper Orthogonal Decomposition for the Nonstationary Navier-Stokes Equations , 2008, SIAM J. Numer. Anal..

[21]  Zhenghui Xie,et al.  A reduced stabilized mixed finite element formulation based on proper orthogonal decomposition for the non‐stationary Navier–Stokes equations , 2011 .

[22]  Ionel M. Navon,et al.  POD-DEIM APPROACH ON DIMENSION REDUCTION OF A MULTI-SPECIES HOST-PARASITOID SYSTEM , 2014 .

[23]  Karsten Urban,et al.  An improved error bound for reduced basis approximation of linear parabolic problems , 2013, Math. Comput..

[24]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[25]  G. Rozza,et al.  On the stability of the reduced basis method for Stokes equations in parametrized domains , 2007 .

[26]  Traian Iliescu,et al.  Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison , 2011, 1106.3585.