Overview of Other Results and Open Problems

  • Olli Mali
  • Pekka Neittaanmäki
  • Sergey Repin
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 32)


This chapter presents an overview of results related to error control methods, which were not considered in previous chapters. In the first part, we discuss possible extensions of the theory exposed in Chaps.  3 and  4 to nonconforming approximations and certain classes of nonlinear problems. Also, we shortly discuss some results related to explicit evaluation of modeling errors. The remaining part of the chapter is devoted to a posteriori estimates of errors in iteration methods. Certainly, the overview is not complete. A posteriori error estimation methods are far from having been fully explored and this subject contains many unsolved problems and open questions, some of which we formulate in the last section.


Variational Inequality Optimal Control Problem Iteration Method Posteriori Error Posteriori Error Estimation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [AD03]
    G. Acosta, R. Duran, An optimal Poincaré inequality in L 1 for convex domains. Proc. Am. Math. Soc. 132(1), 195–202 (2003) MathSciNetGoogle Scholar
  2. [AH83]
    G. Alefeld, J. Herzberger, Introduction to Interval Computations. Computer Science and Applied Mathematics (Academic Press, New York, 1983). Translated from the German by Jon Rokne MATHGoogle Scholar
  3. [AMN11]
    G. Akrivis, C. Makridakis, R.H. Nochetto, Galerkin and Runge-Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence. Numer. Math. 118(3), 429–456 (2011) MathSciNetMATHGoogle Scholar
  4. [Axe94]
    O. Axelsson, Iterative Solution Methods (Cambridge University Press, Cambridge, 1994) MATHGoogle Scholar
  5. [BBC94]
    R. Barrett, M. Berry, T.F. Chan et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods (Society for Industrial and Applied Mathematics, Philadelphia, 1994) Google Scholar
  6. [BBRR07]
    R. Becker, M. Braack, R. Rannacher, T. Richter, Mesh and model adaptivity for flow problems, in Reactive Flows, Diffusion and Transport, ed. by W. Jager, R. Rannacher, J. Warnatz (Springer, Berlin, 2007), pp. 47–75 Google Scholar
  7. [BE03]
    M. Braack, A. Ern, A posteriori control of modeling errors and discretization errors. Multiscale Model. Simul. 1(2), 221–238 (2003) MathSciNetMATHGoogle Scholar
  8. [BHS08]
    D. Braess, R.W.H. Hoppe, J. Schöberl, A posteriori estimators for obstacle problems by the hypercircle method. Comput. Vis. Sci. 11(4–6), 351–362 (2008) MathSciNetGoogle Scholar
  9. [BKR00]
    R. Becker, H. Kapp, R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J. Control Optim. 39(1), 113–132 (2000) MathSciNetMATHGoogle Scholar
  10. [BMR08]
    M. Bildhauer, M. Fuchs, S. Repin, Duality based a posteriori error estimates for higher order variational inequalities with power growth functionals. Ann. Acad. Sci. Fenn., Ser. A 1 Math. 33(2), 475–490 (2008) MathSciNetMATHGoogle Scholar
  11. [BMR12]
    S. Bartels, A. Mielke, T. Roubíček, Quasi-static small-strain plasticity in the limit of vanishing hardening and its numerical approximation. SIAM J. Numer. Anal. 50(2), 951–976 (2012) MathSciNetMATHGoogle Scholar
  12. [BR00]
    H.M. Buss, S.I. Repin, A posteriori error estimates for boundary value problems with obstacles, in Numerical Mathematics and Advanced Applications, ed. by P. Neittaanmäki, T. Tiihonen, P. Tarvainen. Jyväskylä, 1999 (World Scientific, River Edge, 2000), pp. 162–170 Google Scholar
  13. [BR01]
    R. Becker, R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10, 1–102 (2001) MathSciNetMATHGoogle Scholar
  14. [BR07]
    M. Bildhauer, S. Repin, Estimates of the deviation from the minimizer for variational problems with power growth functionals. J. Math. Sci. (N.Y.) 143(2), 2845–2856 (2007) MathSciNetGoogle Scholar
  15. [Bra05]
    D. Braess, A posteriori error estimators for obstacle problems—another look. Numer. Math. 101(3), 421–451 (2005) MathSciNetGoogle Scholar
  16. [But03]
    J.C. Butcher, Numerical Methods for Ordinary Differential Equations (Wiley, Chichester, 2003) MATHGoogle Scholar
  17. [CDNR09]
    S. Cochez-Dhondt, S. Nicaise, S. Repin, A posteriori error estimates for finite volume approximations. Math. Model. Nat. Phenom. 4(1), 106–122 (2009) MathSciNetMATHGoogle Scholar
  18. [CDP97]
    A. Charbonneau, K. Dossou, R. Pierre, A residual-based a posteriori error estimator for Ciarlet-Raviart formulation of the first biharmonic problem. Numer. Methods Partial Differ. Equ. 13(1), 93–111 (1997) MathSciNetMATHGoogle Scholar
  19. [CL55]
    E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955) MATHGoogle Scholar
  20. [CN00]
    Z. Chen, R.H. Nochetto, Rezidual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. 84(4), 527–548 (2000) MathSciNetMATHGoogle Scholar
  21. [Col64]
    L. Collatz, Funktionanalysis und Numerische Mathematik (Springer, Berlin, 1964) Google Scholar
  22. [Deu04]
    P. Deuflhard, Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms (Springer, Berlin, 2004) Google Scholar
  23. [Dob03]
    M. Dobrowolski, On the LBB constant on stretched domains. Math. Nachr. 254/255, 64–67 (2003) MathSciNetGoogle Scholar
  24. [EKK+95]
    T. Eirola, A.M. Krasnosel’skii, M.A. Krasnosel’skii, N.A. Kuznersov, O. Nevanlinna, Incomplete corrections in nonlinear problems. Nonlinear Anal. 25(7), 717–728 (1995) MathSciNetMATHGoogle Scholar
  25. [ES00]
    H.W. Engl, O. Scherzer, Convergence rates results for iterative methods for solving nonlinear ill-posed problems, in Surveys on Solution Methods for Inverse Problems, ed. by D. Colton, H.W. Engl, A.K. Louis, J.R. McLaughlin, W. Rundell (Springer, Vienna, 2000), pp. 7–34 Google Scholar
  26. [EW94]
    S.C. Eisenstat, H.F. Walker, Globally convergent inexact Newton methods. SIAM J. Optim. 4(2), 393–422 (1994) MathSciNetMATHGoogle Scholar
  27. [Fal74]
    R.S. Falk, Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28, 963–971 (1974) MathSciNetMATHGoogle Scholar
  28. [FNP09]
    M. Farhloul, S. Nicaise, L. Paquet, A priori and a posteriori error estimations for the dual mixed finite element method of the Navier-Stokes problem. Numer. Methods Partial Differ. Equ. 25(4), 843–869 (2009) MathSciNetMATHGoogle Scholar
  29. [FR06]
    M. Fuchs, S. Repin, A posteriori error estimates of functional type for variational problems related to generalized Newtonian fluids. Math. Methods Appl. Sci. 29(18), 2225–2244 (2006) MathSciNetMATHGoogle Scholar
  30. [FR10]
    M. Fuchs, 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(9), 1136–1147 (2010) MathSciNetMATHGoogle Scholar
  31. [Fro04b]
    M. Frolov, Reliable control over approximation errors by functional type a posteriori estimates. Ph.D. thesis, Jyväskylä Studies in Computing 44, University of Jyväskylä, 2004 Google Scholar
  32. [GH08]
    A. Günther, M. Hinze, A posteriori error control of a state constrained elliptic control problem. J. Numer. Math. 16(4), 307–322 (2008) MathSciNetMATHGoogle Scholar
  33. [GHR07]
    A. Gaevskaya, R.W.H. Hoppe, S. Repin, Functional approach to a posteriori error estimation for elliptic optimal control problems with distributed control. J. Math. Sci. (N.Y.) 144(6), 4535–4547 (2007) MathSciNetMATHGoogle Scholar
  34. [GL96]
    G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd edn. (Johns Hopkins University Press, Baltimore, 1996) MATHGoogle Scholar
  35. [GR05]
    A. Gaevskaya, S. Repin, A posteriori error estimates for approximate solutions of linear parabolic problems. Differ. Equ. 41(7), 970–983 (2005) MathSciNetMATHGoogle Scholar
  36. [GT74]
    W.B. Gragg, R.A. Tapia, Optimal error bounds for the Newton-Kantorovich theorem. SIAM J. Numer. Anal. 11, 10–13 (1974) MathSciNetMATHGoogle Scholar
  37. [GT77]
    D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 1977) MATHGoogle Scholar
  38. [GTG76]
    E.G. Geisler, A.A. Tal, D.P. Garg, On a-posteriori error bounds for the solution of ordinary nonlinear differential equations, in Computers and Mathematics with Applications, ed. by E.Y. Rodin (Pergamon, Oxford, 1976), pp. 407–416 Google Scholar
  39. [Hay79]
    Y. Hayashi, On a posteriori error estimation in the numerical solution of systems of ordinary differential equations. Hiroshima Math. J. 9(1), 201–243 (1979) MathSciNetMATHGoogle Scholar
  40. [HH08]
    M. Hintermüller, R.H.W. Hoppe, Goal-oriented adaptivity in control constrained optimal control of partial differential equations. SIAM J. Control Optim. 47(4), 1721–1743 (2008) MathSciNetMATHGoogle Scholar
  41. [HHIK08]
    M. Hintermüller, R.H.W. Hoppe, Y. Iliash, M. Kieweg, An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM Control Optim. Calc. Var. 14(3), 540–560 (2008) MathSciNetMATHGoogle Scholar
  42. [HK94]
    R.H.W. Hoppe, R. Kornhuber, Adaptive multilevel methods for obstacle problems. SIAM J. Numer. Anal. 31(2), 301–323 (1994) MathSciNetMATHGoogle Scholar
  43. [HK10]
    R.H.W. Hoppe, M. Kieweg, Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems. Comput. Optim. Appl. 46(3), 511–533 (2010) MathSciNetMATHGoogle Scholar
  44. [HN96]
    J. Haslinger, P. Neittaanmäki, Finite Element Approximation for Optimal Shape, Material and Topology Design, 2nd edn. (Wiley, Chichester, 1996) MATHGoogle Scholar
  45. [HNW93]
    E. Hairer, S.P. Nørsett, G. Wanner, Solving Ordinary Differential Equations. I: Nonstiff Problems, 2nd edn. Springer Series in Computational Mathematics, vol. 8 (Springer, Berlin, 1993) MATHGoogle Scholar
  46. [HW96]
    E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. (Springer, Berlin, 1996) MATHGoogle Scholar
  47. [JSV10]
    P. Jiranek, Z. Strakos, M. Vohralik, A posteriori error estimates including algebraic error and stopping criteria for iterative solvers. SIAM J. Sci. Comput. 32(3), 1567–1590 (2010) MathSciNetMATHGoogle Scholar
  48. [KF75]
    A.N. Kolmogorov, S.V. Fomin, Introductory Real Analysis (Dover, New York, 1975) Google Scholar
  49. [Kol11]
    L.Yu. Kolotilina, On Ostrowski’s disk theorem and lower bounds for the smallest eigenvalues and singular values. J. Math. Sci. (N.Y.) 176(1), 68–77 (2011) MathSciNetGoogle Scholar
  50. [Kor96]
    R. Kornhuber, A posteriori error estimates for elliptic variational inequalities. Comput. Math. Appl. 31(1), 49–60 (1996) MathSciNetMATHGoogle Scholar
  51. [Leo94]
    A.S. Leonov, Some a posteriori stopping rules for iterative methods for solving linear ill-posed problems. Comput. Math. Math. Phys. 34(1), 121–126 (1994) MathSciNetGoogle Scholar
  52. [Lin94]
    E. Lindelöf, Sur l’application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre, in Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, vol. 116, ed. by M.J. Bertrand (Impremerie Gauthier-Villars et Fils, Paris, 1894), pp. 454–457 Google Scholar
  53. [LRT09]
    R. Lazarov, S. Repin, S. Tomar, Functional a posteriori error estimates for discontinuous Galerkin method. Numer. Methods Partial Differ. Equ. 25(4), 952–971 (2009) MathSciNetMATHGoogle Scholar
  54. [Mey92]
    P. Meyer, A unifying theorem on Newton’s method. Numer. Funct. Anal. Optim. 13(5–6), 463–473 (1992) MathSciNetMATHGoogle Scholar
  55. [MM80]
    P. Mosolov, P. Myasnikov, Mechanics of Rigid Plastic Bodies (Nauka, Moscow, 1980). In Russian Google Scholar
  56. [MN06]
    C. Makridakis, R.H. Nochetto, A posteriori error analysis for higher order dissipative methods for evolution problems. Numer. Math. 104(4), 489–514 (2006) MathSciNetMATHGoogle Scholar
  57. [MNR12]
    S. Matculevich, P. Neittaanmäki, S. Repin, Guaranteed error bounds for a class of Picard-Lindelof iteration methods, in Numerical Methods for Differential Equations, Optimization, and Technological Problems, ed. by S. Repin, T. Tiihonen, T. Tuovinen (Springer, Berlin, 2012), pp. 151–168 Google Scholar
  58. [Mor89]
    I. Moret, A Kantorovich-type theorem for inexact Newton methods. Numer. Funct. Anal. Optim. 10(3–4), 351–365 (1989) MathSciNetMATHGoogle Scholar
  59. [MR08]
    O. Mali, S. Repin, Estimates of the indeterminacy set for elliptic boundary-value problems with uncertain data. J. Math. Sci. 150(1), 1869–1874 (2008) MathSciNetGoogle Scholar
  60. [MR10]
    O. Mali, S. Repin, Two-sided estimates of the solution set for the reaction-diffusion problem with uncertain data, in Issue Dedicated to the Jubilee of Prof. R. Glowinski, ed. by W. Fitzgibbon, Y. Kuznetsov, P. Neittaanmäki, J. Periaux, O. Pironneau. Comput. Methods Appl. Sci., vol. 15 (Springer, New York, 2010), pp. 183–198 Google Scholar
  61. [MRV09]
    D. Meidner, R. Rannacher, J. Vihharev, Goal-oriented error control of the iterative solution of finite element equations. J. Numer. Math. 17(2), 143–172 (2009) MathSciNetMATHGoogle Scholar
  62. [Nev89a]
    O. Nevanlinna, Remarks on Picard-Lindelöf iteration I. BIT Numer. Math. 29(2), 328–346 (1989) MathSciNetMATHGoogle Scholar
  63. [Nev89b]
    O. Nevanlinna, Remarks on Picard-Lindelöf iteration II. BIT Numer. Math. 29(3), 535–562 (1989) MathSciNetMATHGoogle Scholar
  64. [NR01]
    P. Neittaanmäki, S. Repin, A posteriori error estimates for boundary-value problems related to the biharmonic operator. East-West J. Numer. Math. 9(2), 157–178 (2001) MathSciNetMATHGoogle Scholar
  65. [NR04]
    P. Neittaanmäki, S. Repin, Reliable Methods for Computer Simulation, Error Control and a Posteriori Estimates (Elsevier, New York, 2004) MATHGoogle Scholar
  66. [NR10b]
    P. Neittaanmäki, S. Repin, A posteriori error majorants for approximations of the evolutionary Stokes problem. J. Numer. Math. 18(2), 119–134 (2010) MathSciNetMATHGoogle Scholar
  67. [NR12]
    A. Nazarov, S. Repin, Exact constants in Poincare type inequalities for functions with zero mean boundary traces. Technical report, 2012. arXiv:1211.2224
  68. [NW03]
    P. Neff, C. Wieners, Comparison of models for finite plasticity: a numerical study. Comput. Vis. Sci. 6(1), 23–35 (2003) MathSciNetMATHGoogle Scholar
  69. [OBN+05]
    J.T. Oden, I. Babushka, F. Nobile, Y. Feng, R. Tempone, Theory and methodology for estimation and control of errors due to modeling, approximation, and uncertainty. Comput. Methods Appl. Mech. Eng. 194(2–5), 195–204 (2005) MATHGoogle Scholar
  70. [OC00]
    M. Olshanskii, E. Chizhonkov, On the best constant in the infsup condition for prolonged rectangular domains. Ž. Vyčisl. Mat. Mat. Fiz. 67(3), 387–396 (2000). In Russian MathSciNetGoogle Scholar
  71. [OPHK01]
    J.T. Oden, S. Prudhomme, D. Hammerand, M. Kuczma, Modeling error and adaptivity in nonlinear continuum mechanics. Comput. Methods Appl. Mech. Eng. 190(49–50), 6663–6684 (2001) MathSciNetMATHGoogle Scholar
  72. [Ort68]
    J.M. Ortega, The Newton-Kantorovich theorem. Am. Math. Mon. 75, 658–660 (1968) MATHGoogle Scholar
  73. [Ost72]
    A. Ostrowski, Les estimations des erreurs a posteriori dans les procédés itératifs. C. R. Acad. Sci. Paris Sér. A-B 275, A275–A278 (1972) MathSciNetGoogle Scholar
  74. [Pay07]
    L.E. Payne, A bound for the optimal constant in an inequality of Ladyzhenskaya and Solonnikov. IMA J. Appl. Math. 72(5), 563–569 (2007) MathSciNetMATHGoogle Scholar
  75. [Pot85]
    F. Potra, Sharp error bounds for a class of Newton-like methods. Libertas Math. 5, 71–84 (1985) MathSciNetMATHGoogle Scholar
  76. [PP80]
    F.-A. Potra, V. Ptak, Sharp error bounds for Newton’s process. Numer. Math. 34(1), 63–72 (1980) MathSciNetMATHGoogle Scholar
  77. [PR09]
    D. Pauly, S. Repin, Functional a posteriori error estimates for elliptic problems in exterior domains. J. Math. Sci. (N.Y.) 163(3), 393–406 (2009) MathSciNetGoogle Scholar
  78. [PRR11]
    D. Pauly, S. Repin, T. Rossi, Estimates for deviations from exact solutions of the Cauchy problem for Maxwell’s equations. Ann. Acad. Sci. Fenn. Math. 36(2), 661–676 (2011) MathSciNetMATHGoogle Scholar
  79. [PTVF07]
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flanner, Numerical Recipes. The Art of Scientific Computing, 3rd edn. (Cambridge University Press, Cambridge, 2007) MATHGoogle Scholar
  80. [PW60]
    L.E. Payne, H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960) MathSciNetMATHGoogle Scholar
  81. [Qn00]
    J. Qi-nian, Error estimates of some Newton-type methods for solving nonlinear inverse problems in Hilbert scales. Inverse Probl. 16, 187–197 (2000) Google Scholar
  82. [Ran00]
    R. Rannacher, The dual-weighted-residual method for error control and mesh adaptation in finite element methods, in The Mathematics of Finite Elements and Applications, X, ed. by J. Whiteman. MAFELAP 1999, Uxbridge (Elsevier, Oxford, 2000), pp. 97–116 Google Scholar
  83. [Ran02]
    R. Rannacher, Adaptive finite element methods for partial differential equations, in Proceedings of the International Congress of Mathematicians, vol. III, ed. by L.I. Tatsien. Beijing, 2002 (Higher Ed. Press, Beijing, 2002), pp. 717–726 Google Scholar
  84. [Rep97a]
    S. Repin, A posteriori error estimation for nonlinear variational problems by duality theory. Zap. Nauč. Semin. POMI 243, 201–214 (1997) Google Scholar
  85. [Rep99a]
    S. Repin, A posteriori estimates for approximate solutions of variational problems with strongly convex functionals. J. Math. Sci. 97(4), 4311–4328 (1999) MathSciNetGoogle Scholar
  86. [Rep00a]
    S. Repin, Estimates of deviations from exact solutions of elliptic variational inequalities. Zap. Nauč. Semin. POMI 271, 188–203 (2000) Google Scholar
  87. [Rep00b]
    S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comput. 69(230), 481–500 (2000) MathSciNetMATHGoogle Scholar
  88. [Rep01a]
    S. Repin, Estimates for errors in two-dimensional models of elasticity theory. J. Math. Sci. (N.Y.) 106(3), 3027–3041 (2001) MathSciNetGoogle Scholar
  89. [Rep02a]
    S. 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(2), 121–133 (2002) MathSciNetMATHGoogle Scholar
  90. [Rep08]
    S. Repin, A Posteriori Error Estimates for Partial Differential Equations (de Gruyter, Berlin, 2008) Google Scholar
  91. [Rep09a]
    S. Repin, Estimates of deviations from exact solutions of initial boundary-value problems for the wave equation. J. Math. Sci. (N.Y.) 159(2), 229–240 (2009) MathSciNetMATHGoogle Scholar
  92. [Rep09b]
    S. Repin, Estimates of deviations from exact solutions of variational inequalities based upon Payne-Weinberger inequality. J. Math. Sci. (N.Y.) 157(6), 874–884 (2009) MathSciNetMATHGoogle Scholar
  93. [Rep12]
    S. Repin, Computable majorants of constants in the Poincare and Friedrichs inequalities. J. Math. Sci. (N.Y.) 186(2), 153–166 (2012) MathSciNetGoogle Scholar
  94. [RR12]
    S. Repin, T. Rossi, On the application of a posteriori estimates of the functional type to quantitative analysis of inverse problems, in Numerical Methods for Differential Equations, Optimization, and Technological Problems, ed. by S. Repin, T. Tiihonen, T. Tuovinen (Springer, Berlin, 2012), pp. 109–120 Google Scholar
  95. [RS10a]
    S. Repin, S. Sauter, Computable estimates of the modeling error related to Kirchhoff-Love plate model. Anal. Appl. 8(4), 409–428 (2010) MathSciNetMATHGoogle Scholar
  96. [RS10b]
    S. Repin, S. Sauter, Estimates of the modeling error for the Kirchhoff-Love plate model. C. R. Math. Acad. Sci. Paris 348(17–18), 1039–1043 (2010) MathSciNetMATHGoogle Scholar
  97. [RSS03]
    S. Repin, S. Sauter, A. Smolianski, A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions. Computing 70(3), 205–233 (2003) MathSciNetMATHGoogle Scholar
  98. [RSS04]
    S. Repin, S. Sauter, A. Smolianski, A posteriori estimation of dimension reduction errors for elliptic problems in thin domains. SIAM J. Numer. Anal. 42(4), 1435–1451 (2004) MathSciNetMATHGoogle Scholar
  99. [RSS12a]
    S. Repin, T. Samrowski, S. Sauter, Combined a posteriori modeling-discretization error estimate for elliptic problems with complicated interfaces. M2AN Math. Model. Numer. Anal. 46(6), 1389–1405 (2012) MathSciNetMATHGoogle Scholar
  100. [RSS12b]
    S. Repin, T. Samrowski, S. Sauter, Two-sided estimates of the modeling error for elliptic homogenization problems. Technical report 12, University of Zurich, 2012 Google Scholar
  101. [RT11]
    S. Repin, S. Tomar, Guaranteed and robust error bounds for nonconforming approximations of elliptic problems. IMA J. Numer. Anal. 31(2), 597–615 (2011) MathSciNetMATHGoogle Scholar
  102. [RV08]
    S. Repin, J. Valdman, Functional a posteriori error estimates for problems with nonlinear boundary conditions. J. Numer. Math. 16(1), 51–81 (2008) MathSciNetMATHGoogle Scholar
  103. [RV09]
    S. Repin, J. Valdman, Functional a posteriori error estimates for incremental models in elasto-plasticity. Cent. Eur. J. Math. 7(3), 506–519 (2009) MathSciNetMATHGoogle Scholar
  104. [RV10]
    R. Rannacher, B. Vexler, Adaptive finite element discretization in PDE-based optimization. GAMM-Mitt. 33(2), 177–193 (2010) MathSciNetMATHGoogle Scholar
  105. [RV12]
    R. Rannacher, J. Vihharev, Balancing discretization and iteration error in finite element a posteriori error analysis, in Numerical Methods for Differential Equations, Optimization, and Technological Problems, ed. by S. Repin, T. Tiihonen, T. Tuovinen (Springer, Berlin, 2012), pp. 85–108 Google Scholar
  106. [RWW10]
    R. Rannacher, A. Westenberger, W. Wollner, Adaptive finite element solution of eigenvalue problems: balancing of discretization and iteration error. J. Numer. Math. 18(4), 303–327 (2010) MathSciNetMATHGoogle Scholar
  107. [RX96]
    S. Repin, L.S. Xanthis, A posteriori error estimation for elastoplastic problems based on duality theory. Comput. Methods Appl. Mech. Eng. 138(1–4), 317–339 (1996) MathSciNetMATHGoogle Scholar
  108. [RX97]
    S. Repin, L.S. Xanthis, A posteriori error estimation for nonlinear variational problems. C. R. Acad. Sci., Sér. 1 Math. 324(10), 1169–1174 (1997) MathSciNetMATHGoogle Scholar
  109. [Saa03]
    Y. Saad, Iterative Methods for Sparse Linear System, 2nd edn. (Society for Industrial and Applied Mathematics, Philadelphia, 2003) Google Scholar
  110. [SG89]
    A. Samarski, A. Gulin, Numerical Methods (Nauka, Moscow, 1989). In Russian Google Scholar
  111. [SO97]
    E. Stein, S. Ohnimus, Coupled model- and solution-adaptivity in the finite element method. Comput. Methods Appl. Mech. Eng. 150(1–4), 327–350 (1997) MathSciNetMATHGoogle Scholar
  112. [SO00]
    M. Schulz, O. Steinbach, A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem. Calcolo 37(2), 79–96 (2000) MathSciNetMATHGoogle Scholar
  113. [Sob50]
    S.L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (Izdt. Leningrad. Gos. Univ., Leningrad, 1950). In Russian Google Scholar
  114. [ST87]
    Y. Stephan, R. Temam, Finite element computation of discontinuous solutions in the perfect plasticity theory, in Computational Plasticity, Part I, II, ed. by D.R.J. Owen, E. Hinton, E. Oñate. Barcelona, 1987 (Pineridge, Swansea, 1987), pp. 243–256 Google Scholar
  115. [SV08]
    M. Schmich, B. Vexler, Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations. SIAM J. Sci. Comput. 30(1), 369–393 (2007/2008) MathSciNetGoogle Scholar
  116. [SW10]
    D. Schötzau, T.P. Wihler, A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations. Numer. Math. 115(3), 475–509 (2010) MathSciNetMATHGoogle Scholar
  117. [TO76]
    K. Tsuruta, K. Ohmori, A posteriori error estimation for Volterra integro-differential equations. Mem. Numer. Math. 3, 33–47 (1976) MathSciNetMATHGoogle Scholar
  118. [TR09]
    S. Tomar, S. Repin, Efficient computable error bounds for discontinuous Galerkin approximations of elliptic problems. J. Comput. Appl. Math. 226(2), 358–369 (2009) MathSciNetMATHGoogle Scholar
  119. [Var62]
    R.S. Varga, Matrix Iterative Analysis (Prentice Hall, Englewood Cliffs, 1962) Google Scholar
  120. [Ver98]
    R. Verfürth, A posteriori error estimates for nonlinear problems. L r(0,T;L ρ(Ω))-error estimates for finite element discretizations of parabolic equations. Math. Comput. 67(224), 1335–1360 (1998) MATHGoogle Scholar
  121. [Ver00]
    V.M. Vergbitskiy, Numerical Methods. Linear Algebra and Non-linear Equations (Nauka, Moscow, 2000) Google Scholar
  122. [Ver03]
    R. Verfürth, A posteriori error estimates for finite element discretizations of the heat equation. Calcolo 40(3), 195–212 (2003) MathSciNetMATHGoogle Scholar
  123. [Ver05]
    R. Verfürth, Robust a posteriori error estimates for nonstationary convection-diffusion equations. SIAM J. Numer. Anal. 43(4), 1783–1802 (2005) MathSciNetMATHGoogle Scholar
  124. [VW08]
    B. Vexler, W. Wollner, Adaptive finite elements for elliptic optimization problems with control constraints. SIAM J. Control Optim. 47(1), 509–534 (2008) MathSciNetMATHGoogle Scholar
  125. [Woh11]
    B. Wohlmuth, Variationally consistent discretization schemes and numerical algorithms for contact problems. Acta Numer. 20, 569–734 (2011) MathSciNetMATHGoogle Scholar
  126. [Yam80]
    T. Yamamoto, Error bounds for computed eigenvalues and eigenvectors. Numer. Math. 34(2), 189–199 (1980) MathSciNetMATHGoogle Scholar
  127. [Yam82]
    T. Yamamoto, Error bounds for computed eigenvalues and eigenvectors. II. Numer. Math. 40(2), 201–206 (1982) MathSciNetMATHGoogle Scholar
  128. [Yam01]
    N. Yamamoto, A simple method for error bounds of the eigenvalues of symmetric matrices. Linear Algebra Appl. 324(1–3), 227–234 (2001) MathSciNetMATHGoogle Scholar
  129. [Zei86]
    E. Zeidler, Nonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems (Springer, New York, 1986) MATHGoogle Scholar
  130. [ZSM05]
    Y. Zhou, R. Shepard, M. Minkoff, Computing eigenvalue bounds for iterative subspace matrix methods. Comput. Phys. Commun. 167(2), 90–102 (2005) MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Olli Mali
    • 1
  • Pekka Neittaanmäki
    • 1
  • Sergey Repin
    • 2
    • 3
  1. 1.Department of Mathematical Information TechnologyUniversity of JyväskyläJyväskyläFinland
  2. 2.Steklov Institute of MathematicsRussian Academy of SciencesSt. PetersburgRussia
  3. 3.University of JyväskyläJyväskyläFinland

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