| Numerical simulation is widely used in control science and engineering.Model dimension reduction can greatly reduce computer running time and improve the efficiency of numerical simulation.In this paper,we use a POD(Proper Orthogonal Decomposition)method to reduce the dimensionality of mixed finite element(MFE)models for unsteady partial differential equations(including unsaturated water equation,unsteady Stokes equation,unsteady Navier-Stokes equation,and unsteady Boussinesq equation)in the control science and engineering so that these MFE models with millions or even tens of millions of unknowns are reduced into the dimension reduction model with only few unknowns.When these dimensionality reduction models are applied to numerical simulations,the computation load and computation time can be greatly reduced and the computational efficiency can be greatly improved.At the same time,the existence,stability,and convergence for the dimensionality reduction solutions are analyzed theoretically,which provides a theoretical basis for practical application.Therefore,this paper has both application value and important theoretical value.The main work of this paper is as follows.(1)An MFE reduced-dimension(MFERD)extrapolation method based on POD method for solution coefficient vectors of the unsaturated water flow problem has been established.The existence,stability,and convergence for the MFERD solutions of soil water content and water flux have been analyzed.The numerical examples of water infiltration and evaporation in soil have been used to verify the correctness of the theoretical results.(2)A Crank-Nicolson(CN)mixed finite spectral element reduced-dimension(CNMFSERD)extrapolation method for the unsteady Stokes equation based on POD method for finite spectral element coefficient vectors has been proposed.The existence,stability,and convergence for the CNMFSERD solutions of fluid velocity have been discussed.The correctness of theoretical results has been verified by the numerical examples of pipe flow with bulge and flow around column.(3)A CN MFERD(CNMFERD)extrapolation method for the unsteady NavierStokes equations based on POD method for the finite element coefficient vectors has been established.The existence,stability,and convergence for the CNMFERD solutions for fluid velocity have been proved.A numerical example of rear step flow problem has been used to verify the correctness of the theoretical results.(4)A CNMFERD extrapolation method for the unsteady Boussinesq equation based on POD method for the finite element coefficient vectors has been established.The existence,stability,and convergence for the CNMFERD solutions have analyzed.A numerical example of flow problem around airfoil has been given to demonstrate the correctness of the theoretical results.The above dimensionality reduction methods can greatly reduce error accumulation,reduce computational load,save computational time,and improve computational efficiency in practical engineering applications.Therefore,this study has important theoretical significance and application Value. |
