Regularity constants of the stokes problem. Application to finite-element methods on curved domains

Mathematical Models and Methods in Applied Sciences

Volumn 15, Issue 3, 2005, Pages 437-470

  Ayuso, B ^a   Garcia Archilla B ^b  

^a UNIVERSIDAD AUTÓNOMA DE MADRID   (Spain)

^b UNIVERSITY OF SEVILLE   (Spain)

Author keywords

Curved domains; Finite element methods; Regularity; Stokes problem

Indexed keywords

EID: 14844348809     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021820250500042X     Document Type: Review

References (29)

1. R. A. Adams, Sobolev Spaces (Academic Press, 1975).
2. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12 (1959) 623-727.
3. T. M. Apostol, Mathematical Analysis, 2nd edn. (Addison-Wesley, 1974).
4. B. Ayuso, The postprocessed mixed finite element method for the Navier-Stokes equations, Ph.D. thesis, Universidad Autónoma de Madrid, 2003.
5. B. Ayuso, B. García-Archilla and J. Novo, Postprocessing mixed finite-element methods for the Navier-Stokes equations, submitted.
6. W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edn. (Academic Press, 1986).
7. N. Yu Bakaev, Maximum norm resolvent estimates for elliptic finite element operators, BIT 41 (2001) 215-239.
8. S. Brenner and R. L. Scott, The Mathematical Theory of Finite Element Methods (Springer-Verlag, 1994).
9. L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Univ. Padova 31 (1961) 308-340.
10. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, Vol. 40 (SIAM, 2002).
11. M. Crouzeix, V. Thomée and L. B. Wahlbin, Error estimates for spatially discrete approximations of semilinear parabolic equations with initial data of low regularity, Math. Comp. 53 (1989) 25-41.
12. N. Dunford and J. T. Schwartz, Linear Operators. Part I: General Theory, with the assistance of William G. Bade and Robert G. Bartle, reprint of the 1958 original. Wiley Classics Library (John Wiley & Sons, 1988).
13. J. de Frutos and J. Novo, Postprocessing the linear finite element method, SIAM J. Numer. Anal. 40 (2002) 805-819.
14. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, reprint of the 1998 edition (Springer-Verlag, 2001).
15. G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems (Springer-Verlag, 1994).
16. q-estimates for the Stokes problem in an exterior domain, Arch. Rational Mech. Anal. 112 (1990) 291-318.
17. B. García-Archilla, J. Novo and E. S. Titi, An approximate inertial manifold approach to postprocessing Galerkin methods for the Navier-Stokes equations, Math. Comp. 68 (1999) 893-911.
18. B. García-Archilla and E. S. Titi, Postprocessing the Galerkin method: The finite-element case, SIAM J. Numer. Anal. 37 (2000) 470-499.
19. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms (Springer-Verlag, 1986).
20. P. M. Gresho and R. L. Sani, Incompressible Flow and the Finite Element Method. Volume Two: Isothermal Laminar Flow (John Wiley & Sons, 1999).
21. P. Hartman, Ordinary Differential Equations, 2nd edn. (SIAM, 2002).
22. C. Johnson, S. Larsson, V. Thomée and L. B. Wahlbin, Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data, Math. Comp. 49 (1987) 331-357.
23. H. Kozono and H. Sohr, New a priori estimates for the Stokes equations in exterior domains, Indiana Univ. Math. J. 40 (1991) 1-28.
24. M. Lenoir, Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM J. Numer. Anal. 23 (1986) 562-580.
25. J. Nitsche, On Dirichlet problems using subspaces with nearly zero boundary conditions, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Proc. Sympos., Univ. Maryland, Baltimore, MD, 1972 (Academic Press, 1972), pp. 603-627.
26. 1-projection into finite-element spaces, Math. Comp. 38 (1982) 1-22.
27. V. Thomée and L. B. Wahlbin, Stability and analyticity in maximum-norm for simplicial Lagrange finite element semidiscretizations of parabolic equations with Dirichlet boundary conditions, Numer. Math. 87 (2000) 373-389.
28. L. B. Wahlbin, Maximum norm error estimates in the finite element method with isoparametric quadratic elements and numerical integration, RAIRO Anal. Numér. 12 (1978) 173-202.
29. L. B. Wahlbin, Superconvergence in Galerkin Finite Element Methods, Lecture Notes in Mathematics, Vol. 1605 (Springer-Verlag, 1995).