Local pointwise a posteriori gradient error bounds for the Stokes equations

Demlow, Alan; Larsson, Stig

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

Local pointwise a posteriori gradient error bounds for the Stokes equations
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by Alan Demlow and Stig Larsson PDF

Math. Comp. 82 (2013), 625-649 Request permission

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.

References

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Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
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
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, DOI 10.1512/iumj.1995.44.2025
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. MR 2218957, DOI 10.1137/040610064
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. MR 2261010, DOI 10.1090/S0025-5718-06-01879-5
A. Demlow and E. Georgoulis, A posteriori error estimates in the maximum norm for discontinuous Galerkin methods, Tech. report, 2010.
Stephen J. Fromm, Potential space estimates for Green potentials in convex domains, Proc. Amer. Math. Soc. 119 (1993), no. 1, 225–233. MR 1156467, DOI 10.1090/S0002-9939-1993-1156467-3
Vivette Girault, Ricardo H. Nochetto, and Ridgway Scott, Stability of the finite element Stokes projection in $W^{1,\infty }$, C. R. Math. Acad. Sci. Paris 338 (2004), no. 12, 957–962 (English, with English and French summaries). MR 2066358, DOI 10.1016/j.crma.2004.04.005
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, DOI 10.1016/j.matpur.2004.09.017
J. Guzmán and D. Leykekhman, Pointwise error estimates of finite element approximations to the Stokes problem on convex polyhedra, Tech. report, 2010.
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, DOI 10.1007/s00211-009-0213-y
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, DOI 10.1007/s00211-008-0141-2
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, DOI 10.1090/surv/085
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, DOI 10.1002/mana.200410340
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, DOI 10.1002/mana.200610513
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, DOI 10.1002/mma.695
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, DOI 10.1090/surv/162
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, DOI 10.1515/9783110848915.525
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, DOI 10.1007/978-1-4419-1343-2_{1}5
—, Personal communication, 2010.
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, DOI 10.1515/9783110812411
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, DOI 10.1090/S0025-5718-1995-1297478-7
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
Erik Daniel Svensson, Computational characterization of mixing in flows, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)–Chalmers Tekniska Hogskola (Sweden). MR 2715918
E. D. Svensson and S. Larsson, Pointwise a posteriori error estimates for the Stokes equations in polyhedral domains, Preprint (2006).