Local pointwise a posteriori gradient error bounds for the Stokes equations

Demlow, Alan; Larsson, Stig

doi:10.1090/S0025-5718-2012-02647-0

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6842(online) ISSN 0025-5718(print)

Previous issue | This issue | All issues (1943–Present)
Articles in press | Recently posted articles | Next article

Local pointwise a posteriori gradient error bounds for the Stokes equations

Authors: Alan Demlow and Stig Larsson
Journal: Math. Comp. 82 (2013), 625-649
MSC (2010): Primary 65N30
Posted: November 27, 2012
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the standard Taylor-Hood finite element method for the stationary Stokes system on polyhedral domains. We prove local a posteriori error estimates for the maximum error in the gradient of the velocity field. Because the gradient of the velocity field blows up near reentrant corners and edges, such local error control is necessary when pointwise control of the gradient error is desirable. Computational examples confirm the utility of our estimates in adaptive codes.

[Ada75] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957 (56 #9247)
[Ama00] Herbert Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glas. Mat. Ser. III 35(55) (2000), no. 1, 161–177. Dedicated to the memory of Branko Najman. MR 1783238 (2001h:46056)
[BS95] Russell M. Brown and Zhongwei Shen, Estimates for the Stokes operator in Lipschitz domains, Indiana Univ. Math. J. 44 (1995), no. 4, 1183–1206. MR 1386766 (97c:35152), http://dx.doi.org/10.1512/iumj.1995.44.2025
[Dem06] Alan Demlow, Localized pointwise a posteriori error estimates for gradients of piecewise linear finite element approximations to second-order quadilinear elliptic problems, SIAM J. Numer. Anal. 44 (2006), no. 2, 494–514 (electronic). MR 2218957 (2007c:65105), http://dx.doi.org/10.1137/040610064
[Dem07] Alan Demlow, Local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems, Math. Comp. 76 (2007), no. 257, 19–42 (electronic). MR 2261010 (2008c:65319), http://dx.doi.org/10.1090/S0025-5718-06-01879-5
[DG10] A. Demlow and E. Georgoulis.
A posteriori error estimates in the maximum norm for discontinuous Galerkin methods.
Technical report, 2010.
[Fro93] Stephen J. Fromm, Potential space estimates for Green potentials in convex domains, Proc. Amer. Math. Soc. 119 (1993), no. 1, 225–233. MR 1156467 (93k:35076), http://dx.doi.org/10.1090/S0002-9939-1993-1156467-3
[GL10] J. Guzmán and D. Leykekhman.
Pointwise error estimates of finite element approximations to the Stokes problem on convex polyhedra.
Technical report, 2010.
[GLRS09] J. Guzmán, D. Leykekhman, J. Rossmann, and A. H. Schatz, Hölder estimates for Green’s functions on convex polyhedral domains and their applications to finite element methods, Numer. Math. 112 (2009), no. 2, 221–243. MR 2495783 (2010a:65237), http://dx.doi.org/10.1007/s00211-009-0213-y
[GNS04] Vivette Girault, Ricardo H. Nochetto, and Ridgway Scott, Stability of the finite element Stokes projection in 𝑊^{1,∞}, C. R. Math. Acad. Sci. Paris 338 (2004), no. 12, 957–962 (English, with English and French summaries). MR 2066358, http://dx.doi.org/10.1016/j.crma.2004.04.005
[GNS05] V. Girault, R. H. Nochetto, and R. Scott, Maximum-norm stability of the finite element Stokes projection, J. Math. Pures Appl. (9) 84 (2005), no. 3, 279–330 (English, with English and French summaries). MR 2121575 (2006j:76087), http://dx.doi.org/10.1016/j.matpur.2004.09.017
[HXZL08] Yinnian He, Jinchao Xu, Aihui Zhou, and Jian Li, Local and parallel finite element algorithms for the Stokes problem, Numer. Math. 109 (2008), no. 3, 415–434. MR 2399151 (2009j:76157), http://dx.doi.org/10.1007/s00211-008-0141-2
[KMR01] V. A. Kozlov, V. G. Maz′ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical Surveys and Monographs, vol. 85, American Mathematical Society, Providence, RI, 2001. MR 1788991 (2001i:35069)
[MR05] V. Maz′ya and J. Rossmann, Pointwise estimates for Green’s kernel of a mixed boundary value problem to the Stokes system in a polyhedral cone, Math. Nachr. 278 (2005), no. 15, 1766–1810. MR 2182091 (2007b:35269), http://dx.doi.org/10.1002/mana.200410340
[MR06] V. G. Maz′ya and J. Rossmann, Schauder estimates for solutions to a mixed boundary value problem for the Stokes system in polyhedral domains, Math. Methods Appl. Sci. 29 (2006), no. 9, 965–1017. MR 2228352 (2007f:35220), http://dx.doi.org/10.1002/mma.695
[MR07] V. Maz’ya and J. Rossmann, 𝐿_{𝑝} estimates of solutions to mixed boundary value problems for the Stokes system in polyhedral domains, Math. Nachr. 280 (2007), no. 7, 751–793. MR 2321139 (2008m:35280), http://dx.doi.org/10.1002/mana.200610513
[MR10] Vladimir Maz’ya and Jürgen Rossmann, Elliptic equations in polyhedral domains, Mathematical Surveys and Monographs, vol. 162, American Mathematical Society, Providence, RI, 2010. MR 2641539 (2011h:35002)
[NP94] Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, de Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387 (95h:35001)
[Ros10a] Jürgen Roßmann, Hölder estimates for Green’s matrix of the Stokes system in convex polyhedra, Around the research of Vladimir Maz’ya. II, Int. Math. Ser. (N. Y.), vol. 12, Springer, New York, 2010, pp. 315–336. MR 2676181 (2012a:35250), http://dx.doi.org/10.1007/978-1-4419-1343-2_15
[Ros10b] J. Rossmann.
Personal communication.
2010.
[RS96] Thomas Runst and Winfried Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, vol. 3, Walter de Gruyter & Co., Berlin, 1996. MR 1419319 (98a:47071)
[SL06] E. D. Svensson and S. Larsson.
Pointwise a posteriori error estimates for the Stokes equations in polyhedral domains.
Preprint, 2006.
[SS05] Alfred Schmidt and Kunibert G. Siebert, Design of adaptive finite element software, Lecture Notes in Computational Science and Engineering, vol. 42, Springer-Verlag, Berlin, 2005. The finite element toolbox ALBERTA; With 1 CD-ROM (Unix/Linux). MR 2127659 (2005i:65003)
[Sve06] Erik Daniel Svensson, Computational characterization of mixing in flows, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–Chalmers Tekniska Hogskola (Sweden). MR 2715918
[SW95] A. H. Schatz and L. B. Wahlbin, Interior maximum-norm estimates for finite element methods. II, Math. Comp. 64 (1995), no. 211, 907–928. MR 1297478 (95j:65143), http://dx.doi.org/10.1090/S0025-5718-1995-1297478-7

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|