Pointwise A Priori Error Estimates for the Stokes Equations in Polyhedral Domains

Abstract. We derive pointwise a posteriori residual-based error estimates for finite element solutions to the Stokes equations in polyhedral domains. The estimates relies on the regularity of the of Stokes equations and provide an upper bound for the pointwise error in the velocity field on polyhedral domains. Whereas the estimates provide upper bounds for the pointwise error in the gradient of the velocity field and the pressure only for a restricted class of polyhedral domains, convex polyhedral domains in R, and polyhedral domains with angles at edges < 3π/4 in R. In the cause of this study we also derive L a posteriori error estimates, generalizing well known L estimates.

[1]  Zhongwei Shen,et al.  Estimates for the Stokes Operator in Lipschitz Domains , 1995 .

[2]  N. S. Barnett,et al.  Private communication , 1969 .

[3]  Ricardo H. Nochetto,et al.  Fully Localized A posteriori Error Estimators and Barrier Sets for Contact Problems , 2004, SIAM J. Numer. Anal..

[4]  Ricardo H. Nochetto,et al.  Pointwise a posteriori error estimates for elliptic problems on highly graded meshes , 1995 .

[5]  Hantaek Bae.  Navier-Stokes equations , 1992 .

[6]  Ricardo H. Nochetto,et al.  Pointwise a Posteriori Error Analysis for an Adaptive Penalty Finite Element Method for the Obstacle Problem , 2001 .

[7]  Ricardo H. Nochetto,et al.  Maximum-norm stability of the finite element Stokes projection , 2005 .

[8]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[9]  Ricardo H. Nochetto,et al.  Weighted inf-sup condition and pointwise error estimates for the Stokes problem , 1990 .

[10]  F. Thomasset.  Finite element methods for Navier-Stokes equations , 1980 .

[11]  Juergen Rossmann,et al.  Lp estimates of solutions tomixed boundary value problems for the Stokes system in polyhedral domains , 2004, math-ph/0412073.

[12]  Kenneth Eriksson,et al.  AN ADAPTIVE FINITE ELEMENT METHOD WITH EFFICIENT MAXIMUM NORM ERROR CONTROL FOR ELLIPTIC PROBLEMS , 1994 .

[13]  Vladimir Maz’ya,et al.  Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations , 2000 .

[14]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[15]  Ricardo H. Nochetto,et al.  Pointwise a posteriori error estimates for monotone semi-linear equations , 2006, Numerische Mathematik.

[16]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[17]  Ricardo H. Nochetto,et al.  Pointwise a posteriori error control for elliptic obstacle problems , 2003, Numerische Mathematik.

[18]  Ricardo G. Durán,et al.  Maximum Norm Error Estimators for Three-Dimensional Elliptic Problems , 1999, SIAM J. Numer. Anal..