### SUPERCONVERGENCE ANALYSIS OF LOW ORDER NONCONFORMING MIXED FINITE ELEMENT METHODS FOR TIME-DEPENDENT NAVIER-STOKES EQUATIONS

Huaijun Yang1, Dongyang Shi2, Qian Liu2

1. 1. School of Mathematics, Zhengzhou University of Aeronautics, Zhengzhou 450046, China;
2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China
• Received:2018-12-02 Revised:2019-05-12 Online:2021-01-15 Published:2021-03-11
• Supported by:
This work is supported by National Natural Science Foundation of China (Nos. 11671369; 11271340).

Huaijun Yang, Dongyang Shi, Qian Liu. SUPERCONVERGENCE ANALYSIS OF LOW ORDER NONCONFORMING MIXED FINITE ELEMENT METHODS FOR TIME-DEPENDENT NAVIER-STOKES EQUATIONS[J]. Journal of Computational Mathematics, 2021, 39(1): 63-80.

In this paper, the superconvergence properties of the time-dependent Navier-Stokes equations are investigated by a low order nonconforming mixed finite element method (MFEM). In terms of the integral identity technique, the superclose error estimates for both the velocity in broken H1-norm and the pressure in L2-norm are first obtained, which play a key role to bound the numerical solution in L-norm. Then the corresponding global superconvergence results are derived through a suitable interpolation postprocessing approach. Finally, some numerical results are provided to demonstrated the theoretical analysis.

CLC Number:

 [1] Xiaocui Li, Xu You. MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2021, 39(1): 130-146. [2] Baiju Zhang, Yan Yang, Minfu Feng. A C0-WEAK GALERKIN FINITE ELEMENT METHOD FOR THE TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS IN STREAM-FUNCTION FORMULATION [J]. Journal of Computational Mathematics, 2020, 38(2): 310-336. [3] Houchao Zhang, Dongyang Shi. SUPERCONVERGENCE ANALYSIS FOR TIME-FRACTIONAL DIFFUSION EQUATIONS WITH NONCONFORMING MIXED FINITE ELEMENT METHOD [J]. Journal of Computational Mathematics, 2019, 37(4): 488-505. [4] Rui Chen, Xiaofeng Yang, Hui Zhang. DECOUPLED, ENERGY STABLE SCHEME FOR HYDRODYNAMIC ALLEN-CAHN PHASE FIELD MOVING CONTACT LINE MODEL [J]. Journal of Computational Mathematics, 2018, 36(5): 661-681. [5] C. Brennecke, A. Linke, C. Merdon, J. Schöberl. OPTIMAL AND PRESSURE-INDEPENDENT L2 VELOCITY ERROR ESTIMATES FOR A MODIFIED CROUZEIX-RAVIART STOKES ELEMENT WITH BDM RECONSTRUCTIONS [J]. Journal of Computational Mathematics, 2015, 33(2): 191-208. [6] Xin He, Maya Neytcheva, Cornelis Vuik. ON PRECONDITIONING OF INCOMPRESSIBLE NON-NEWTONIAN FLOW PROBLEMS [J]. Journal of Computational Mathematics, 2015, 33(1): 33-58. [7] Xin He, Maya Neytcheva. PRECONDITIONING THE INCOMPRESSIBLE NAVIER-STOKESEQUATIONS WITH VARIABLE VISCOSITY [J]. Journal of Computational Mathematics, 2012, 30(5): 461-482. [8] Minfu Feng, Yanhong Bai, Yinnian He, Yanmei Qin. A NEW STABILIZED SUBGRID EDDY VISCOSITY METHOD BASED ON PRESSURE PROJECTION AND EXTRAPOLATED TRAPEZOIDAL RULE FOR THE TRANSIENT NAVIER-STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2011, 29(4): 415-440. [9] M. Gunzburger, A. Labovsky. EFFECTS OF APPROXIMATE DECONVOLUTION MODELS ON THE SOLUTION OF THE STOCHASTIC NAVIER-STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2011, 29(2): 131-140. [10] Houde Han Ming Yan. A Mixed Finite Element Method on a Staggered Mesh for Navier-Stokes Equations [J]. Journal of Computational Mathematics, 2008, 26(6): 816-824. [11] Junping Wang, Xiaoshen Wang, Xiu Ye . Finite Element Methods for the Navier-Stokes Equations by $H(\rm{div})$ Elements [J]. Journal of Computational Mathematics, 2008, 26(3): 410-436. [12] Yin Nian HE, Huan Ling MIAO, Chun Feng REN. A TWO-LEVEL FINITE ELEMENT GALERKIN METHOD FOR THE NONSTATIONARY NAVIER-STOKES EQUATIONS II: TIME DISCRETIZATION [J]. Journal of Computational Mathematics, 2004, 22(1): 33-54. [13] Chun Feng REN , Yi Chen MA. A TWO-GRID METHOD FOR THE STEADY PENALIZED NAVIER-STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2004, 22(1): 101-112. [14] Yin Nian HE. A TWO-LEVEL FINITE ELEMENT GALERKIN METHOD FOR THE NONSTATIONARY NAVIER-STOKES EQUATIONS I: SPATIAL DISCRETIZATION [J]. Journal of Computational Mathematics, 2004, 22(1): 21-32. [15] Yin Nian HE, Yan Ren HOU, Li Quan MEI. GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS [J]. Journal of Computational Mathematics, 2001, 19(6): 607-616.
Viewed
Full text

Abstract