Estimates of deviations from exact solutions for boundary-value problems with incompressibility condition
HTML articles powered by AMS MathViewer
- by
S. Repin
Translated by: the author - St. Petersburg Math. J. 16 (2005), 837-862
- DOI: https://doi.org/10.1090/S1061-0022-05-00882-4
- Published electronically: September 21, 2005
- PDF | Request permission
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–Babuška–Brezzi condition.References
- 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. Ladyzhenskaya, Kraevye zadachi matematicheskoĭ fiziki, Izdat. “Nauka”, Moscow, 1973 (Russian). MR 0599579
- O. A. Ladyženskaja, Modifications of the Navier-Stokes equations for large gradients of the velocities, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7 (1968), 126–154 (Russian). MR 0241832
- 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
- O. A. Ladyzhenskaya and N. N. Ural′tseva, Lineĭnye i kvazilineĭnye uravneniya èllipticheskogo tipa, Izdat. “Nauka”, Moscow, 1973 (Russian). Second edition, revised. MR 0509265
- 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, Ž. Vyčisl. Mat i Mat. Fiz. 9 (1969), 1102–1120 (Russian). MR 295599
- 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
- S. I. Repin, A posteriori error estimates for approximate solutions to variational problems with strongly convex functionals, J. Math. Sci. (New York) 97 (1999), no. 4, 4311–4328. Problems of mathematical physics and function theory. MR 1788233, DOI 10.1007/BF02365047
- S. I. Repin, Estimates for errors in two-dimensional models of elasticity theory, J. Math. Sci. (New York) 106 (2001), no. 3, 3027–3041. Function theory and phase transitions. MR 1906032, DOI 10.1023/A:1011323722683
- S. I. Repin, A posteriori estimates for the accuracy of variational methods for problems with nonconvex functionals, Algebra i Analiz 11 (1999), no. 4, 151–182 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 11 (2000), no. 4, 651–672. MR 1713937
- S. I. Repin, Two-sided estimates of deviation from exact solutions of uniformly elliptic equations, Proceedings of the St. Petersburg Mathematical Society, Vol. IX, Amer. Math. Soc. Transl. Ser. 2, vol. 209, Amer. Math. Soc., Providence, RI, 2003, pp. 143–171. MR 2018375, DOI 10.1090/trans2/209/06
- G. Astarita and G. Marrucci, Principles of non-Newtonian fluid mechanics, McGraw-Hill, London, 1974.
- Mark Ainsworth and J. Tinsley Oden, A posteriori error estimators for the Stokes and Oseen equations, SIAM J. Numer. Anal. 34 (1997), no. 1, 228–245. MR 1445736, DOI 10.1137/S0036142994264092
- Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
- 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 799997, DOI 10.1007/BF02576171
- Ivo Babuška, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1972/73), 179–192. MR 359352, DOI 10.1007/BF01436561
- I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), no. 4, 736–754. MR 483395, DOI 10.1137/0715049
- Ivo Babuška and Theofanis Strouboulis, The finite element method and its reliability, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2001. MR 1857191
- Randolph E. Bank and Bruno D. Welfert, A posteriori error estimates for the Stokes problem, SIAM J. Numer. Anal. 28 (1991), no. 3, 591–623. MR 1098409, DOI 10.1137/0728033
- Alfredo Bermúdez, Ricardo Durán, and Rodolfo Rodríguez, Finite element analysis of compressible and incompressible fluid-solid systems, Math. Comp. 67 (1998), no. 221, 111–136. MR 1434937, DOI 10.1090/S0025-5718-98-00901-6
- James 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. Dedicated to Jim Douglas, Jr. on the occasion of his 75th birthday. MR 1977631, DOI 10.1142/S0218202503002544
- 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
- Franco Brezzi and Jim Douglas Jr., Stabilized mixed methods for the Stokes problem, Numer. Math. 53 (1988), no. 1-2, 225–235. MR 946377, DOI 10.1007/BF01395886
- Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
- Carsten Carstensen and Sören Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM, Math. Comp. 71 (2002), no. 239, 945–969. MR 1898741, DOI 10.1090/S0025-5718-02-01402-3
- 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. Math. 8 (2000), no. 3, 153–175. MR 1807259
- Carsten Carstensen and Stefan A. Funken, A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems, Math. Comp. 70 (2001), no. 236, 1353–1381. MR 1836908, DOI 10.1090/S0025-5718-00-01264-3
- Alexandre Joel Chorin, On the convergence of discrete approximations to the Navier-Stokes equations, Math. Comp. 23 (1969), 341–353. MR 242393, DOI 10.1090/S0025-5718-1969-0242393-5
- Ph. Clément, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9 (1975), no. R-2, 77–84 (English, with French summary). MR 0400739
- Enzo Dari, Ricardo Durán, and Claudio Padra, Error estimators for nonconforming finite element approximations of the Stokes problem, Math. Comp. 64 (1995), no. 211, 1017–1033. MR 1284666, DOI 10.1090/S0025-5718-1995-1284666-9
- Weinan E and Jian-Guo Liu, Projection method. I. Convergence and numerical boundary layers, SIAM J. Numer. Anal. 32 (1995), no. 4, 1017–1057. MR 1342281, DOI 10.1137/0732047
- Ivar Ekeland and Roger Temam, Convex analysis and variational problems, Studies in Mathematics and its Applications, Vol. 1, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. Translated from the French. MR 0463994
- Martin Fuchs and Gregory Seregin, Variational methods for problems from plasticity theory and for generalized Newtonian fluids, Lecture Notes in Mathematics, vol. 1749, Springer-Verlag, Berlin, 2000. MR 1810507, DOI 10.1007/BFb0103751
- V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR 548867
- John G. Heywood and Rolf Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982), no. 2, 275–311. MR 650052, DOI 10.1137/0719018
- John G. Heywood and Rolf Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization, SIAM J. Numer. Anal. 27 (1990), no. 2, 353–384. MR 1043610, DOI 10.1137/0727022
- C. Johnson and R. Rannacher, On error control in computational fluid dynamics, Preprint no. 1994-07, Dept. Math. Chalmers Univ. of Technology, Goteborg, 1994.
- Claes Johnson, Rolf Rannacher, and Mats Boman, Numerics and hydrodynamic stability: toward error control in computational fluid dynamics, SIAM J. Numer. Anal. 32 (1995), no. 4, 1058–1079. MR 1342282, DOI 10.1137/0732048
- Georgij M. Kobelkov and Maxim A. Olshanskii, Effective preconditioning of Uzawa type schemes for a generalized Stokes problem, Numer. Math. 86 (2000), no. 3, 443–470. MR 1785417, DOI 10.1007/s002110000160
- O. A. Ladyženskaja, Certain nonlinear problems of the theory of continuous media, Proc. Internat. Congr. Math. (Moscow, 1966) Izdat. “Mir”, Moscow, 1968, pp. 560–573 (Russian). MR 0239291
- J. Málek, J. Nečas, M. Rokyta, and M. Růžička, Weak and measure-valued solutions to evolutionary PDEs, Applied Mathematics and Mathematical Computation, vol. 13, Chapman & Hall, London, 1996. MR 1409366, DOI 10.1007/978-1-4899-6824-1
- P. P. Mosolov and V. P. Myasnikov, Mekhanika zhestkoplasticheskikh sred, “Nauka”, Moscow, 1981 (Russian). MR 641694
- J. Tinsley Oden, Weihan Wu, and Mark Ainsworth, An a posteriori error estimate for finite element approximations of the Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 111 (1994), no. 1-2, 185–202. MR 1259620, DOI 10.1016/0045-7825(94)90045-0
- Claudio Padra, A posteriori error estimators for nonconforming approximation of some quasi-Newtonian flows, SIAM J. Numer. Anal. 34 (1997), no. 4, 1600–1615. MR 1461798, DOI 10.1137/S0036142994278322
- Rolf Rannacher, Numerical analysis of the Navier-Stokes equations, Proceedings of ISNA ’92—International Symposium on Numerical Analysis, Part I (Prague, 1992), 1993, pp. 361–380. MR 1228513
- Rolf Rannacher, Finite element methods for the incompressible Navier-Stokes equations, Fundamental directions in mathematical fluid mechanics, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2000, pp. 191–293. MR 1799399
- S. I. Repin, A posteriori error estimation for nonlinear variational problems by duality theory, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 243 (1997), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funktsiĭ. 28, 201–214, 342 (English, with English and Russian summaries); English transl., J. Math. Sci. (New York) 99 (2000), no. 1, 927–935. MR 1629741, DOI 10.1007/BF02673600
- 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
- Sergey I. Repin, A unified approach to a posteriori error estimation based on duality error majorants, Math. Comput. Simulation 50 (1999), no. 1-4, 305–321. Modelling ’98 (Prague). MR 1717590, DOI 10.1016/S0378-4754(99)00081-6
- S. I. Repin, Estimates of deviations from exact solutions of elliptic variational inequalities, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 271 (2000), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 31, 188–203, 317 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 115 (2003), no. 6, 2811–2819. MR 1810617, DOI 10.1023/A:1023378021130
- 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 of deviations for generalized Newtonian fluids, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288 (2002), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 32, 178–203, 273 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 123 (2004), no. 6, 4621–4636. MR 1923550, DOI 10.1023/B:JOTH.0000041479.59584.10
- Sergey Repin, 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), no. 2, 121–133 (English, with English and Italian summaries). MR 1949485
- 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, Computing 70 (2003), no. 3, 205–233. MR 2011610, DOI 10.1007/s00607-003-0013-7
- Sergey Repin, Stefan Sauter, and Anton Smolianski, A posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions, Proceedings of the 10th International Congress on Computational and Applied Mathematics (ICCAM-2002), 2004, pp. 601–612. MR 2056902, DOI 10.1016/S0377-0427(03)00491-6
- Jie 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, DOI 10.1090/S0025-5718-96-00750-8
- Roger Temam, Navier-Stokes equations, Revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979. Theory and numerical analysis; With an appendix by F. Thomasset. MR 603444
- R. Verfürth, A review of a posteriori error estimation and adaptive mesh–refinement techniques, Wiley; Teubner, New York, 1996.
- R. Verfürth, A posteriori error estimators for the Stokes equations, Numer. Math. 55 (1989), no. 3, 309–325. MR 993474, DOI 10.1007/BF01390056
- R. Verfürth, A posteriori error estimators for the Stokes equations. II. Nonconforming discretizations, Numer. Math. 60 (1991), no. 2, 235–249. MR 1133581, DOI 10.1007/BF01385723
- Lars B. Wahlbin, Superconvergence in Galerkin finite element methods, Lecture Notes in Mathematics, vol. 1605, Springer-Verlag, Berlin, 1995. MR 1439050, DOI 10.1007/BFb0096835
- Miloš Zlámal, Some superconvergence results in the finite element method, Mathematical aspects of finite element methods (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
- 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), no. 2, 337–357. MR 875306, DOI 10.1002/nme.1620240206
Bibliographic Information
- S. Repin
- Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
- Email: repin@pdmi.ras.ru
- 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)
- © Copyright 2005 American Mathematical Society
- 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
- MathSciNet review: 2106670
Dedicated: Dedicated to the memory of O. A. Ladyzhenskaya