Abstract
We consider a finite element method for the penalty formulation of the time dependent Navier-Stokes equations. Usually the improper choice of the finite element space will lead that the error estimate (inversely) depends on the penalty parameter \({\epsilon}\). We use the classical P 1 nonconforming finite element space for the spatial discretization. Optimal \({\epsilon}\)-uniform error estimations for both velocity and pressure are obtained.
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Andreas P.: Projection and Quasi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations. B. G. Teubner, Stuttgart (1997)
Bramble J.H., Hilbert S.R.: Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7, 113–124 (1970)
Brefort B., Ghidaglia J.M., Temam R.: Attractors for the penalized Navier-Stokes equations. SIAM J. Math. Anal. 19, 1–21 (1988)
Brenner S.C., Sung L.-Y.: Linear finite element methods for planar linear elasticity. Math. Comput. 59, 321–338 (1992)
Brenner S.C., Scott L.R.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)
Brezzi, F., Pitkaranta, J.: On the stabilization of finite element approximation of the Stokes problems. In: Efficient Solution of Elliptic system, Notes on Numerical Fluids Mechanics, vol. 10, pp. 11–19 (1984)
Chorin A.J.: Numerical solution of the Navier-Stokes equations. Math. Comput. 23, 745–762 (1969)
Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Crouzeix M., Raviart P.A.: Comforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO R-3, 33–75 (1973)
Ern, A., Guermond, J.L.: Theory and Practice of finite elements. Applied Mathematical Sciences Series. Springer, New York (2004)
Girault V., Raviart P.A.: Finite Element Method for Navier-Stokes Equations: Theory and Algorithm. Springer, Berlin (1987)
He Y.: Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes equations. Math. Comput. 74, 1201–1216 (2005)
He Y., Li J., Yang X.: Two-level penalized finite element methods for the stationary Navier-Stoke equations. Int. J. Inf. Syst. Sci. 2, 131–143 (2006)
Heywood J.G., Rannacher R.: Finite element approximation of the nonstationary Navier-Stokes problem, I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275–311 (1982)
Hughes T.J.R., Liu W.T., Brooks A.J.: Finite element analysis of incompressible viscous flows by the penalty function formulation. J. Comput. Phys. 30, 1–60 (1979)
Lin P.: A sequential regularization methods for time-dependent incompressibel Navier-Stokes equations. SIAM J. Numer. Anal. 34, 1051–1071 (1997)
Lin P., Liu J.G., Lu X.L.: Long time numerical solution of the Navier-Stokes equations based on a sequential regularization formulation. SIAM J. Sci. Comput. 31, 398–419 (2008)
Lin P., Liu C.: Simulation of singularity dynamics in liquid crystal flows: a C 0 finite element approach. J. Comput. Phys. 215, 348–362 (2006)
Lin P., Liu C., Zhang H.: An energy law preserving C 0 finite element scheme for simulating the kinematic effects in liquid crystal flow dynamics. J. Comput. Phys. 227, 1411–1427 (2007)
Lu, X.: Error Analysis for incompressible viscous flow based on a sequential regularization formulation. Ph.D. Dissertation (2006)
Lu X.L., Lin P., Liu J.G.: Analysis of a sequential regularization method for the unsteady Navier-Stokes equations. Math. Comput. 77, 1467–1494 (2008)
Payne L.E., Weinberger H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)
Shen J.: On error estimates of the penalty method for the unsteady Navier-Stokes equations. SIAM J. Numer. Anal. 32, 386–403 (1995)
Temam R.: Navier-Stokes Equations. North-Holland, Amsterdam (1977)
Temam R.: Une méthode d’approximation de la solution des équations de Navier-Stokes. Bull. Soc. Math. France 98, 115–152 (1968)
Temam R.: Sure L’approximation de la solution des equations de Navier-Stokes par la méthods des pas fractionnaires II. Arch. Ration. Mech. Anal. 33, 377–385 (1969)
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Lu, X., Lin, P. Error estimate of the P 1 nonconforming finite element method for the penalized unsteady Navier-Stokes equations. Numer. Math. 115, 261–287 (2010). https://doi.org/10.1007/s00211-009-0277-8
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DOI: https://doi.org/10.1007/s00211-009-0277-8