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Estimates of deviations from exact solutions for boundary-value problems with incompressibility condition


Author: S. Repin
Translated by: the author
Original publication: Algebra i Analiz, tom 16 (2004), nomer 5.
Journal: St. Petersburg Math. J. 16 (2005), 837-862
MSC (2000): Primary 35J50, 65N15, 74G45
DOI: https://doi.org/10.1090/S1061-0022-05-00882-4
Published electronically: September 21, 2005
MathSciNet review: 2106670
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Abstract: Methods of estimating the difference between exact and approximate solutions are considered for boundary-value problems in spaces of solenoidal functions. The estimates obtained apply to any functions in the energy space of the respective problem, and their computation requires solving only finite-dimensional problems. In the paper, two different methods are considered: one involves variational formulations and duality theory, and in the other, estimates are obtained from the integral identities that define generalized solutions of the problems in question. It is shown that estimates of deviations from an exact solution must include an additional penalty term with a factor determined by the constant in the Ladyzhenskaya-Babuska-Brezzi condition.


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  • 1. O. A. Ladyzhenskaya, Mathematical problems in the dynamics of a viscous incompressible fluid, 2nd ed., ``Nauka'', Moscow, 1970; English transl. of 1st ed., The mathematical theory of viscous incompressible flow, Gordon and Breach Sci. Publishers, New York-London, 1963. MR 0155093 (27:5034a)
  • 2. -, Boundary value problems of mathematical physics, ``Nauka'', Moscow, 1973; English transl., Appl. Math. Sci., vol. 49, Springer-Verlag, New York-Berlin, 1985. MR 0599579 (58:29032); MR 0793735 (87f:35001)
  • 3. -, Modifications of the Navier-Stokes equations for large velocity gradients, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 126-154; English transl., Sem. Math. Steklov Math. Inst., Leningrad 7 (1970), 57-69. MR 0241832 (39:3169)
  • 4. O. A. Ladyzhenskaya and V. A. Solonnikov, Some problems of vector analysis and generalized formulations of boundary-value problems for the Navier-Stokes equations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 59 (1976), 81-116; English transl., J. Soviet Math. 10 (1978), no. 2, 257-286. MR 0467031 (57:6900)
  • 5. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear equations of elliptic type, 2nd ed., ``Nauka'', Moscow, 1973; English transl. of 1st ed., Acad. Press, New York-London, 1968. MR 0509265 (58:23009); MR 0244627 (39:5941)
  • 6. L. A. Oganesjan and L. A. Ruhovec, An investigation of the rate of convergence of variation-difference schemes for second order elliptic equations in a two-dimensional region with smooth boundary, Zh. Vychisl. Mat. i Mat. Fiz. 9 (1969), no. 5, 1102-1120. (Russian) MR 0295599 (45:4665)
  • 7. M. A. Ol'shanskii 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; English transl., Math. Notes 67 (2000), no. 3-4, 325-332. MR 1779472 (2001g:76020)
  • 8. S. I. Repin, A posteriori error estimates for approximate solutions to variational problems with strongly convex functionals, Probl. Mat. Anal., No. 17, S.-Peterburg. Univ., St. Petersburg, 1997, pp. 199-226; English transl., J. Math. Sci. 97 (1999), no. 4, 4311-4328. MR 1788233 (2001i:49053)
  • 9. -, Estimates for errors in two-dimensional models of elasticity theory, Probl. Mat. Anal., No. 22, S.-Peterburg. Univ., St. Petersburg, 2001, pp. 178-196; English transl., J. Math. Sci. 106 (2001), no. 3, 3027-3041. MR 1906032 (2003d:74028)
  • 10. -, A posteriori estimates of the accuracy of variational methods for problems with nonconvex functionals, Algebra i Analiz 11 (1999), no. 4, 151-182; English transl., St. Petersburg Math. J. 11 (2000), no. 4, 651-672. MR 1713937 (2001f:49039)
  • 11. -, Two-sided estimates of deviation from exact solutions of uniformly elliptic equations, Trudy S.-Peterburg. Mat. Obshch. 9 (2001), 148-179; English transl., Amer. Math. Soc. Transl. Ser. 2, vol. 209, Amer. Math. Soc., Providence, RI, 2003, pp. 143-171. MR 2018375 (2004k:35081)
  • 12. G. Astarita and G. Marrucci, Principles of non-Newtonian fluid mechanics, McGraw-Hill, London, 1974.
  • 13. M. Ainsworth and J. T. Oden, A posteriori error estimators for the Stokes and Oseen equations, SIAM J. Numer. Anal. 34 (1997), no. 1, 228-245. MR 1445736 (98c:65180)
  • 14. -, A posteriori error estimation in finite element analysis, Wiley, New York, 2000. MR 1885308 (2003b:65001)
  • 15. D. N. Arnold, F. Brezzi, and M. Fortin, A stable finite element for the Stokes equations, Calcolo 21 (1984), no. 4, 337-344 (1985). MR 0799997 (86m:65136)
  • 16. I. Babuska, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1973), 179-192. MR 0359352 (50:11806)
  • 17. I. Babuska and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), 736-754. MR 0483395 (58:3400)
  • 18. I. Babuska and T. Strouboulis, The finite element method and its reliability, Clarendon Press, New York, 2001. MR 1857191 (2002k:65001)
  • 19. R. E. Bank and B. D. Welfert, A posteriori error estimates for the Stokes problem, SIAM J. Numer. Anal. 28 (1991), no. 3, 591-623. MR 1098409 (92a:65284)
  • 20. A. Bermúdez, R. Durán, and R. Rodriguez, Finite element analysis of compressible and incompressible fluid-solid systems, Math. Comp. 67 (1998), no. 221, 111-136. MR 1434937 (98c:73073)
  • 21. J. H. Bramble, A proof of the inf-sup condition for the Stokes equations on Lipschitz domains, Math. Models Methods Appl. Sci. 13 (2003), no. 3, 361-371. MR 1977631 (2004d:35197)
  • 22. F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, RAIRO Sér. Rouge Anal. Numér. 8 (1974), no. R-2, 129-151. MR 0365287 (51:1540)
  • 23. F. Brezzi and J. Douglas, Stabilized mixed methods for the Stokes problem, Numer. Math. 53 (1988), no. 1-2, 225-235. MR 0946377 (89g:65138)
  • 24. F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Springer Ser. Comput. Math., vol. 15, Springer-Verlag, New York, 1991. MR 1115205 (92d:65187)
  • 25. C. Carstensen and S. Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. . Low order conforming, nonconforming, and mixed FEM, Math. Comp. 71 (2002), no. 239, 945-969. MR 1898741 (2003e:65212)
  • 26. C. Carstensen and S. A. Funken, Constants in Clément-interpolation error and residual based a posteriori error estimates in finite element methods, East-West J. Numer. Anal. 8 (2000), no. 3, 153-175. MR 1807259 (2002a:65173)
  • 27. -, A posteriori error control in low-order finite element discretizations of incompressible stationary flow problems, Math. Comp. 70 (2001), no. 236, 1353-1381 (electronic). MR 1836908 (2002f:65157)
  • 28. A. J. Chorin, On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comp. 23 (1969), 341-353. MR 0242393 (39:3724)
  • 29. Ph. Clément, Approximation by finite element functions using local regularization, RAIRO Sér. Rouge Anal. Numér. 9 (1975), no. R-2, 77-84. MR 0400739 (53:4569)
  • 30. E. Dari, R. Durán, and C. Padra, Error estimators for nonconforming finite element approximations of the Stokes problem, Math. Comp. 64 (1995), no. 211, 1017-1033. MR 1284666 (95j:65136)
  • 31. Weinan E and J.-G. Liu, Projection method. . Convergence and numerical boundary layers, SIAM J. Numer. Anal. 32 (1995), no. 4, 1017-1057. MR 1342281 (96e:65061)
  • 32. I. Ekeland and R. Temam, Convex analysis and variational problems, Stud. Math. Appl., vol. 1, North-Holland, Amsterdam-Oxford, 1976. MR 0463994 (57:3931b)
  • 33. M. Fuchs and G. A. Seregin, Variational methods for problems from plasticity theory and for generalized Newtonian fluids, Lecture Notes in Math., vol. 1749, Springer-Verlag, Berlin, 2000. MR 1810507 (2001k:74043)
  • 34. V. Girault and P. A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Math., vol. 749, Springer-Verlag, Berlin-New York, 1979. MR 0548867 (83b:65122)
  • 35. J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. . Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982), 275-311. MR 0650052 (83d:65260)
  • 36. -, Finite-element approximation of the nonstationary Navier-Stokes problem. . Error analysis for second-order time discretization, SIAM J. Numer. Anal. 27 (1990), 353-384. MR 1043610 (92c:65133)
  • 37. C. Johnson and R. Rannacher, On error control in computational fluid dynamics, Preprint no. 1994-07, Dept. Math. Chalmers Univ. of Technology, Goteborg, 1994.
  • 38. C. Johnson, R. Rannacher, and M. Boman, Numerics and hydrodynamic stability: toward error control in computational fluid dynamics, SIAM J. Numer. Anal. 32 (1995), 1058-1079. MR 1342282 (96j:76089)
  • 39. G. M. Kobelkov and M. Olshanskii, Effective preconditioning of Uzawa type schemes for a generalized Stokes problem, Numer. Math. 86 (2000), 443-470. MR 1785417 (2001j:65168)
  • 40. O. A. Ladyzhenskaya, Certain nonlinear problems of the theory of continuous media, Proc. Internat. Congr. Math. (Moscow, 1966), ``Mir'', Moscow, 1968, pp. 560-573. (Russian). (A preliminary version of this article has appeared in Amer. Math. Soc. Transl. Ser. 2, vol. 70, Amer. Math. Soc., Providence, RI, 1968, pp. 73-89.) MR 0239291 (39:648)
  • 41. J. Málek, J. Necas, M. Rokyta, and M. R $\overset{\circ}{{u}}$zicka, Weak and measure-valued solutions to evolutionary PDES, Appl. Math. Math. Comput., vol. 13, Chapman and Hall, London, 1996. MR 1409366 (97g:35002)
  • 42. P. P. Mosolov and V. P. Myasnikov, Mechanics of rigid plastic media, ``Nauka'', Moscow, 1981. (Russian) MR 0641694 (84e:73029)
  • 43. J. T. Oden, W. Wu, and M. Ainsworth, A posteriori error estimate for finite element approximations of the Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 111 (1994), 185-202. MR 1259620 (95e:76054)
  • 44. C. Padra, A posteriori error estimators for nonconforming approximation of some quasi-Newtonian flows, SIAM J. Numer. Anal. 34 (1997), 1600-1615. MR 1461798 (98h:65050)
  • 45. R. Rannacher, Numerical analysis of the Navier-Stokes equations, Appl. Math. 38 (1993), 361-380. MR 1228513 (94h:65101)
  • 46. -, Finite element methods for the incompressible Navier-Stokes equations, Fundamental Directions in Mathematical Fluid Mechanics (P. Galdi, J. H. Heywood, R Rannacher, eds.), Birkhäuser, Basel, 2000, pp. 191-293. MR 1799399 (2001m:76061)
  • 47. S. Repin, A posteriori error estimation for nonlinear variational problems by duality theory, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 243 (1997), 201-214; English transl., J. Math. Sci. 99 (2000), no. 1, 927-935. MR 1629741 (99e:49049)
  • 48. -, A posteriori error estimation for variational problems with uniformly convex functionals, Math. Comp. 69 (2000), no. 230, 481-500. MR 1681096 (2000i:49046)
  • 49. -, A unified approach to a posteriori error estimation based on duality error majorants, Math. Comput. Simulation 50 (1999), 305-321. MR 1717590 (2000h:65085)
  • 50. -, Estimates of deviations from exact solutions of elliptic variational inequalities, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), 188-203; English transl., J. Math. Sci. 115 (2003), no. 6, 2811-2819. MR 1810617 (2001m:35134)
  • 51. -, A posteriori estimates for the Stokes problem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 259 (1999), 195-211; English transl., J. Math. Sci. 109 (2002), no. 5, 1950-1964. MR 1754364 (2001c:35193)
  • 52. -, Estimates of deviations for generalized Newtonian fluids, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288 (2002), 178-203; English transl., J. Math. Sci. (N.Y.) 123 (2004), no. 6, 4621-4636. MR 1923550 (2003g:35178)
  • 53. -, Estimates of deviations from exact solutions of initial-boundary value problem for the heat equation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13 (2002), 121-133. MR 1949485 (2004b:65133)
  • 54. S. Repin, S. Sauter, and A. Smolianski, A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions, Preprint no. 03-2002, Univ. of Zurich, Inst. of Math., 2002; Computing 70 (2003), no. 3, 205-233. MR 2011610 (2004j:65168)
  • 55. -, A posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions, J. Comput. Appl. Math. 164-165 (2004), 601-612. MR 2056902
  • 56. J. Shen, On error estimates of the projection methods for the Navier-Stokes equations: second-order schemes, Math. Comp. 65 (1996), no. 215, 1039-1065. MR 1348047 (96j:65091)
  • 57. R. Temam, Navier-Stokes equations. Theory and numerical analysis, Stud. Math. Appl., vol. 2, North-Holland, Amsterdam-New York, 1979. MR 0603444 (82b:35133)
  • 58. R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley; Teubner, New York, 1996.
  • 59. -, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), 309-325. MR 0993474 (90d:65187)
  • 60. -, A posteriori error estimators for the Stokes equations. . Nonconforming discretizations, Numer. Math. 60 (1991), 235-249. MR 1133581 (92j:65189)
  • 61. L. B. Wahlbin, Superconvergence in Galerkin finite element methods, Lecture Notes in Math., vol. 1605, Springer-Verlag, Berlin, 1995. MR 1439050 (98j:65083)
  • 62. M. Zlámal, Some superconvergence results in the finite element method, Mathematical Aspects of Finite Element Methods (A. Dold, B. Eckmann, eds.) (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), Lecture Notes in Math., vol. 606, Springer, Berlin, 1977, pp. 353-362. MR 0488863 (58:8365)
  • 63. O. C. Zienkiewicz and J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis, Internat. J. Numer. Methods Engrg. 24 (1987), 337-357. MR 0875306 (87m:73055)

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Additional Information

DOI: https://doi.org/10.1090/S1061-0022-05-00882-4
Keywords: Boundary-value problems with incompressibility condition, \emph{a posteriori} error estimates, Stokes problem
Received by editor(s): December 10, 2003
Published electronically: September 21, 2005
Additional Notes: This work was supported by the Civilian Research and Development Foundation (grant no. RU-M1-2596-ST-04) and by the Russian Ministry of Education (grant no. E02-1.0-55)
Dedicated: Dedicated to the memory of O. A. Ladyzhenskaya
Article copyright: © Copyright 2005 American Mathematical Society

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