This work concerns the analysis and finite element approximations of the evolutionary Stokes equations, with inhomogeneous boundary and divergence data. The proposed weak formulation can be viewed as an attempt to develop the parabolic analog of the well known saddle point theory for elliptic problems. Several results concerning the analysis and finite element approximations are presented. The key feature of the weak formulation under consideration is the treatment of Dirichlet boundary conditions within the Lagrange multiplier framework.
Mots clés : Evolutionary stokes equations, inhomogeneous boundary and divergence data, error estimates, finite element approximations, lagrange multipliers, saddle point formulations
@article{M2AN_2017__51_4_1501_0,
author = {Chrysafinos, Konstantinos and Hou, L. Steven},
title = {Analysis and approximations of the evolutionary {Stokes} equations with inhomogeneous boundary and divergence data using a parabolic saddle point formulation},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1501--1526},
publisher = {EDP-Sciences},
volume = {51},
number = {4},
year = {2017},
doi = {10.1051/m2an/2016070},
mrnumber = {3702422},
zbl = {1469.76062},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2016070/}
}
TY - JOUR AU - Chrysafinos, Konstantinos AU - Hou, L. Steven TI - Analysis and approximations of the evolutionary Stokes equations with inhomogeneous boundary and divergence data using a parabolic saddle point formulation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1501 EP - 1526 VL - 51 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016070/ DO - 10.1051/m2an/2016070 LA - en ID - M2AN_2017__51_4_1501_0 ER -
%0 Journal Article %A Chrysafinos, Konstantinos %A Hou, L. Steven %T Analysis and approximations of the evolutionary Stokes equations with inhomogeneous boundary and divergence data using a parabolic saddle point formulation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1501-1526 %V 51 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016070/ %R 10.1051/m2an/2016070 %G en %F M2AN_2017__51_4_1501_0
Chrysafinos, Konstantinos; Hou, L. Steven. Analysis and approximations of the evolutionary Stokes equations with inhomogeneous boundary and divergence data using a parabolic saddle point formulation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1501-1526. doi : 10.1051/m2an/2016070. http://www.numdam.org/articles/10.1051/m2an/2016070/
, The finite element method with Lagrange multipliers. Numer. Math. 20 (1973) 179–192. | DOI | MR | Zbl
and , On mixed finite element method with Lagrange multipliers. Numer. Method Partial Differ. Equ 19 (2003) 192–210. | DOI | MR | Zbl
, and , A penalized approach for solving a parabolic equation with nonsmooth Dirischlet boundary conditions. Asymptot. Anal. 34 (2003) 121–136. | MR | Zbl
, On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers. RAIRO Anal. Numer. 8 (1974) 129–151. | Numdam | MR | Zbl
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag (1991). | MR | Zbl
and , Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim. 45 (2006) 1586–1611. | DOI | MR | Zbl
and , Semi-discrete error estimates for the evolutionary Stokes equations with inhomogeneous Dirischlet boundary data. Comput. Math. Appl. 73 (2017) 1684–1696. | DOI | MR | Zbl
P.G. Ciarlet, The finite element method for elliptic problems. Classics in Applied Math. SIAM (2002). | MR | Zbl
M. Dauge, Elliptic boundary value problems on corner domains. Vol. 1341 of Lect. Notes Math., Springer Verlag, Berlin (1988). | MR | Zbl
, and , A new class of weak solutions of the Navier–Stokes equations with nonhomogeneous data. J. Math. Fluid Mech. 8 (2006) 423–444. | DOI | MR | Zbl
, Control problems and theorems concerning the unique solvability of a mixed boundary value problem for the three-dimensional Navier–Stokes and Euler equations. Math USSR Sb. 43 (1982) 281–307. | DOI | MR | Zbl
, and , Boundary value problems and optimal boundary control for the Navier–Stokes equations, SIAM J. Control Optim. 36 (1998) 852–894. | DOI | MR | Zbl
, and , Inhomogeneous boundary value problems for the three-dimensional evolutionary Navier–Stokes eqiations. J. Math. Fluid Mech. 4 (2002) 45–75. | DOI | MR | Zbl
, and , Trace theorems for three-dimensional, time dependent solenoidal vector fields and their applications, Trans. Amer. Math. Soc. 354 (2002) 1079–1116. | DOI | MR | Zbl
V. Girault and P.-A. Raviart, Finite Element Methods for Navier–Stokes Equations. Springer Verlag (1986). | MR | Zbl
R. Glowinski, Numerical Methods for Nonlinear Variational Problems. Springer Verlag (1984). | MR | Zbl
P. Grisvard, Elliptic problems in Non-smooth domains. Pitman, Boston (1985). | MR | Zbl
, and , Space-time variational saddle point formulations of Stokes and Navier–Stokes equations. ESAIM: M2AN 48 (2014) 875–894. | DOI | Numdam | MR | Zbl
M.D. Gunzburger, Perspectives in flow control and optimization, in: Advances in Design and Control. SIAM Philadelphia (2003). | MR | Zbl
and , Treating Inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses. SIAM. J. Numer. Anal. 29 (1992) 390–424. | DOI | MR | Zbl
and , 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 (1982) 275–311. | DOI | MR | Zbl
Error estimates for semidiscrete finite element approximations of the Stokes equations under minimal regularity assumptions. J. Sci. Comput. 16 (2001) 287–317. | DOI | MR | Zbl
O. Ladyzheskaya, The Mathematical Theory of Viscous Incompressible flow. Gordon and Breach (1969). | MR
, Quelques résultats d’existence dans des équations aux dérivées partielles non linéaires. Bull. Soc. Math. France 87 (1959) 245–273. | DOI | Numdam | MR | Zbl
J.-L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires. Dunod (1968). | MR | Zbl
, Sur une méthode pour résoudre le équations aux dérivées partielles du type elliptique voisine de la variationnnelle. Ann. Scuola Norm. Sup. Pisa 16 (1962) 305–326. | Numdam | MR | Zbl
, Boundary subspaces for the finite element method with Lagrange multipliers. Numer. Math. 33 (1979) 273–289. | DOI | MR | Zbl
, Boundary feedback stabilization of the two dimensional Navier–Stokes equations. SIAM J. Control Optim. 45 (2006) 790–828. | DOI | MR | Zbl
, Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6 (2007) 921–951. | DOI | Numdam | MR | Zbl
, Stokes and Navier–Stokes equations with an nonhomogeneous divergence condition. Discrete Contin. Dyn. Syst. Ser. B. 14 (2010) 1537–1564. | MR | Zbl
and , Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483-493. | DOI | MR | Zbl
, Estimates for solutions of nonstationary Navier-Stokes equaions. J. Soviet. Math. 8 (1977) 213–317. | DOI | Zbl
R. Temam, Navier Stokes Equations. North Holland, Amsterdam (1979). | MR | Zbl
, Finite element approximation of incompressible Navier–Stokes equations with slip bopundary condition. Numer. Math. 50 (1987) 697–721. | DOI | MR | Zbl
E. Zeidler, Nonlinear functional analysis and its application-Linear monotone operators. Springer Verlag New York (1990). | MR | Zbl
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