A posteriori estimates for the stationary Stokes problem in exterior domains

Pauly, D.; Repin, S.

doi:10.1090/spmj/1613

A posteriori estimates for the stationary Stokes problem in exterior domains
HTML articles powered by AMS MathViewer

by D. Pauly and S. Repin

St. Petersburg Math. J. 31 (2020), 533-555

DOI: https://doi.org/10.1090/spmj/1613

Published electronically: April 30, 2020

PDF | Request permission

Abstract:

This paper is concerned with the analysis of the inf-sup condition arising in the stationary Stokes problem in exterior domains and applications to the derivation of computable bounds for the distance between the exact solution of the exterior Stokes problem and a certain approximation (which may be of a rather general form). In the first part, guaranteed bounds are deduced for the constant in the stability lemma associated with the exterior domain. These bounds depend only on known constants and the stability constant related to bounded domains that arise after suitable truncations of the unbounded domains. The lemma in question implies computable estimates of the distance to the set of divergence free fields defined in exterior domains. Such estimates are crucial for the derivation of computable majorants of the difference between the exact solution of the Stokes problem in exterior domains and an approximation from the admissible (energy) class of functions satisfying the Dirichlet boundary condition but not necessarily divergence free (solenoidal). Estimates of this type are often called a posteriori estimates of functional type. The constant in the stability lemma (or equivalently in the inf-sup or LBB condition) serves as a penalty factor at the term that controls violations of the divergence free condition. In the last part of the paper, similar estimates are deduced for the distance to the exact solution for nonconforming approximations, i.e., for those that may violate some continuity and boundary conditions. The case where the dimension of the domain equals $2$ requires a special consideration because the corresponding weighted spaces differ from those natural for the dimension $3$ (or larger). This special case is briefly discussed at the end of the paper where similar estimates are deduced for the distance to the exact solution of the exterior Stokes problem.

References

Ivo Babuška and A. K. Aziz, Survey lectures on the mathematical foundations of the finite element method, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR 0421106
Sebastian Bauer, Dirk Pauly, and Michael Schomburg, The Maxwell compactness property in bounded weak Lipschitz domains with mixed boundary conditions, SIAM J. Math. Anal. 48 (2016), no. 4, 2912–2943. MR 3542004, DOI 10.1137/16M1065951
F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with French summary). MR 365287
S. Cochez-Dhondt, S. Nicaise, and S. Repin, A posteriori error estimates for finite volume approximations, Math. Model. Nat. Phenom. 4 (2009), no. 1, 106–122. MR 2483555, DOI 10.1051/mmnp/20094105
Martin Costabel, A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains, Math. Methods Appl. Sci. 12 (1990), no. 4, 365–368. MR 1048563, DOI 10.1002/mma.1670120406
Martin Costabel and Monique Dauge, On the inequalities of Babuška-Aziz, Friedrichs and Horgan-Payne, Arch. Ration. Mech. Anal. 217 (2015), no. 3, 873–898. MR 3356990, DOI 10.1007/s00205-015-0845-2
M. Fuchs and S. Repin, Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids, Math. Methods Appl. Sci. 33 (2010), no. 9, 1136–1147. MR 2668900, DOI 10.1002/mma.1242
Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. Linearized steady problems. MR 1284205, DOI 10.1007/978-1-4612-5364-8
Vivette Girault and Adélia Sequeira, A well-posed problem for the exterior Stokes equations in two and three dimensions, Arch. Rational Mech. Anal. 114 (1991), no. 4, 313–333. MR 1100798, DOI 10.1007/BF00376137
C. O. Horgan and L. E. Payne, On inequalities of Korn, Friedrichs and Babuška-Aziz, Arch. Rational Mech. Anal. 82 (1983), no. 2, 165–179. MR 687553, DOI 10.1007/BF00250935
M. Kessler, Die Ladyzhenskaya-Konstante in der numerischen Behandlung von Strömungsproblemen, Dissertat. Doktorgrades, Bayerischen Julius-Maximilians-Universität, Würzburg, 2000.
Peter Kuhn and Dirk Pauly, Regularity results for generalized electro-magnetic problems, Analysis (Munich) 30 (2010), no. 3, 225–252. MR 2676208, DOI 10.1524/anly.2010.1024
O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Revised English edition, Gordon and Breach Science Publishers, New York-London, 1963. Translated from the Russian by Richard A. Silverman. MR 0155093
O. A. Ladyženskaja and V. A. Solonnikov, Some problems of vector analysis, and generalized formulations of boundary value problems for the Navier-Stokes equation, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 59 (1976), 81–116, 256 (Russian, with English summary). Boundary value problems of mathematical physics and related questions in the theory of functions, 9. MR 0467031
Raytcho Lazarov, Sergey Repin, and Satyendra K. Tomar, Functional a posteriori error estimates for discontinuous Galerkin approximations of elliptic problems, Numer. Methods Partial Differential Equations 25 (2009), no. 4, 952–971. MR 2526991, DOI 10.1002/num.20386
Rolf Leis, Initial-boundary value problems in mathematical physics, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR 841971, DOI 10.1007/978-3-663-10649-4
Olli Mali, Pekka Neittaanmäki, and Sergey Repin, Accuracy verification methods, Computational Methods in Applied Sciences, vol. 32, Springer, Dordrecht, 2014. Theory and algorithms. MR 3136124, DOI 10.1007/978-94-007-7581-7
A. Mikhaylov and S. Repin, Estimates of deviations from exact solution of the Stokes problem in the vorticity-velocity-pressure formulation, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 397 (2011), no. Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 42, 73–88, 173; English transl., J. Math. Sci. (N.Y.) 185 (2012), no. 5, 698–706. MR 2870109, DOI 10.1007/s10958-012-0953-6
S. G. Mikhlin, Variational methods in mathematical physics, A Pergamon Press Book, The Macmillan Company, New York, 1964. Translated by T. Boddington; editorial introduction by L. I. G. Chambers. MR 0172493
Solomon G. Mikhlin, Constants in some inequalities of analysis, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1986. Translated from the Russian by Reinhard Lehmann. MR 853915
S. A. Nazarov and K. Pileckas, On steady Stokes and Navier-Stokes problems with zero velocity at infinity in a three-dimensional exterior domain, J. Math. Kyoto Univ. 40 (2000), no. 3, 475–492. MR 1794517, DOI 10.1215/kjm/1250517677
Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584
P. Neittaanmäki and S. Repin, Reliable methods for computer simulation, Studies in Mathematics and its Applications, vol. 33, Elsevier Science B.V., Amsterdam, 2004. Error control and a posteriori estimates. MR 2095603
P. Neittaanmäki and S. Repin, A posteriori error majorants for approximations of the evolutionary Stokes problem, J. Numer. Math. 18 (2010), no. 2, 119–134. MR 2657641, DOI 10.1515/JNUM.2010.005
M. A. Ol′shanskiĭ and E. V. Chizhonkov, On the best constant in the inf-sup condition for elongated rectangular domains, Mat. Zametki 67 (2000), no. 3, 387–396 (Russian, with Russian summary); English transl., Math. Notes 67 (2000), no. 3-4, 325–332. MR 1779472, DOI 10.1007/BF02676669
Dirk Pauly, Generalized electro-magneto statics in nonsmooth exterior domains, Analysis (Munich) 27 (2007), no. 4, 425–464. MR 2373665, DOI 10.1524/anly.2007.27.4.425
Dirk Pauly and Sergei Repin, Functional a posteriori error estimates for elliptic problems in exterior domains, J. Math. Sci. (N.Y.) 162 (2009), no. 3, 393–406. Problems in mathematical analysis. No. 42. MR 2839012, DOI 10.1007/s10958-009-9643-4
—, The stationary Stokes problem in exterior domains: estimates of the distance to solenoidal fields and functional a posteriori error estimates, https://arxiv.org/abs/1810.12555, 2018.
L. E. Payne, A bound for the optimal constant in an inequality of Ladyzhenskaya and Solonnikov, IMA J. Appl. Math. 72 (2007), no. 5, 563–569. MR 2361570, DOI 10.1093/imamat/hxm028
Rainer Picard, Zur Theorie der harmonischen Differentialformen, Manuscripta Math. 27 (1979), no. 1, 31–45 (German, with English summary). MR 524976, DOI 10.1007/BF01297736
R. Picard, Randwertaufgaben in der verallgemeinerten Potentialtheorie, Math. Methods Appl. Sci. 3 (1981), no. 2, 218–228 (German, with English summary). MR 657293, DOI 10.1002/mma.1670030116
Rainer Picard, On the boundary value problems of electro- and magnetostatics, Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), no. 1-2, 165–174. MR 667134, DOI 10.1017/S0308210500020023
W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5 (1947), 241–269. MR 25902, DOI 10.1090/S0033-569X-1947-25902-8
Sergey I. Repin, A posteriori error estimation for variational problems with uniformly convex functionals, Math. Comp. 69 (2000), no. 230, 481–500. MR 1681096, DOI 10.1090/S0025-5718-99-01190-4
S. I. Repin, A posteriori estimates for the Stokes problem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 259 (1999), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 30, 195–211, 299 (English, with English and Russian summaries); English transl., J. Math. Sci. (New York) 109 (2002), no. 5, 1950–1964. MR 1754364, DOI 10.1023/A:1014400626472
S. I. Repin, Estimates for deviations from exact solutions of some boundary value problems with the incompressibility condition, Algebra i Analiz 16 (2004), no. 5, 124–161 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 5, 837–862. MR 2106670, DOI 10.1090/S1061-0022-05-00882-4
Sergey Repin, A posteriori estimates for partial differential equations, Radon Series on Computational and Applied Mathematics, vol. 4, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. MR 2458008, DOI 10.1515/9783110203042
S. Repin, Estimates of the distance to the set of divergence free fields, J. Math. Sci. (N.Y.) 210 (2015), no. 6, 822–834. MR 3407795, DOI 10.1007/s10958-015-2593-0
Sergey Repin, Estimates of the distance to the set of solenoidal vector fields and applications to a posteriori error control, Comput. Methods Appl. Math. 15 (2015), no. 4, 515–530. MR 3403448, DOI 10.1515/cmam-2015-0024
S. Repin, Localized forms of the LBB condition and a posteriori estimates for incompressible media problems, Math. Comput. Simulation 145 (2018), 156–170. MR 3725807, DOI 10.1016/j.matcom.2016.05.004
S. Repin and R. Stenberg, A posteriori estimates for a generalized Stokes problem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 362 (2008), no. Kraevye Zacachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 39, 272–302, 367 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 159 (2009), no. 4, 541–558. MR 2760557, DOI 10.1007/s10958-009-9460-9
Sergey Repin, Stanislav Sysala, and Jaroslav Haslinger, Computable majorants of the limit load in Hencky’s plasticity problems, Comput. Math. Appl. 75 (2018), no. 1, 199–217. MR 3758699, DOI 10.1016/j.camwa.2017.09.007
J. Saranen and K. J. Witsch, Exterior boundary value problems for elliptic equations, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), no. 1, 3–42. MR 698835, DOI 10.5186/aasfm.1983.0821
G. Stoyan, Towards discrete Velte decompositions and narrow bounds for inf-sup constants, Comput. Math. Appl. 38 (1999), no. 7-8, 243–261. MR 1713178, DOI 10.1016/S0898-1221(99)00254-0
S. K. Tomar and S. I. Repin, Efficient computable error bounds for discontinuous Galerkin approximations of elliptic problems, J. Comput. Appl. Math. 226 (2009), no. 2, 358–369. MR 2502931, DOI 10.1016/j.cam.2008.08.015
R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), no. 3, 309–325. MR 993474, DOI 10.1007/BF01390056
—, A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley-Teubner, Stuttgart, 1996.
Hermann Weyl, The method of orthogonal projection in potential theory, Duke Math. J. 7 (1940), 411–444. MR 3331