Numerische Mathematik

, Volume 134, Issue 1, pp 1–25 | Cite as

Justification of the saturation assumption

  • C. CarstensenEmail author
  • D. Gallistl
  • J. Gedicke


The saturation assumption is widely used in computational science and engineering, usually without any rigorous theoretical justification and even despite of counterexamples for some coarse meshes known in the mathematical literature. On the other hand, there is overwhelming numerical evidence at least in an asymptotic regime for the validity of the saturation. In the generalized form, the assumption states, for any \(0<\varepsilon \le 1\), that
$$\begin{aligned} ||| u - {\hat{U}} |||^2 \le (1- \varepsilon /C) ||| u - U |||^2+ \varepsilon \mathrm{osc}^2(f,\mathcal {N}) \end{aligned}$$
for the exact solution u and the first-order conforming finite element solution U (resp. \({\hat{U}}\)) of the Poisson model problem with respect to a regular triangulation \(\mathcal {T}\) (resp. \({\hat{\mathcal {T}}}\)) and its uniform refinement \({\hat{\mathcal {T}}}\) within the class \(\mathbb {T}\) of admissible triangulations. The point is that the patch-oriented oscillations \(\mathrm{osc}(f,\mathcal {N})\) vanish for constant right-hand sides \(f\equiv 1\) and may be of higher order for smooth f, while the strong reduction factor \((1- \varepsilon /C)<1\) involves some universal constant C which exclusively depends on the set of admissible triangulations and so on the initial triangulation only. This paper proves the inequality (SA) for the energy norms of the errors for any admissible triangulation \(\mathcal {T}\) in \(\mathbb {T}\) up to computable pathological situations characterized by failing the weak saturation test (WS). This computational test (WS) for some triangulation \(\mathcal {T}\) states that the solutions U and \({\hat{U}}\) do not coincide for the constant right-hand side \(f\equiv 1\). The set of possible counterexamples is characterized as \(\mathcal {T}\) with no interior node or exactly one interior node which is the vertex of all triangles and \({\hat{\mathcal {T}}}\) is a particular uniform bisec3 refinement. In particular, the strong saturation assumption holds for all triangulations with more than one degree of freedom. The weak saturation test (WS) is only required for zero or one degree of freedom and gives a definite outcome with O(1) operations. The only counterexamples known so far are regular n-polygons. The paper also discusses a generalization to linear elliptic second-order PDEs with small convection to prove that saturation is somehow generic and fails only in very particular situations characterised by (WS).

Mathematics Subject Classification

65N15 65N30 


  1. 1.
    Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. In: Pure and Applied Mathematics (New York). Wiley-Interscience (John Wiley & Sons), New York (2000)Google Scholar
  2. 2.
    Bank, R.E., Smith, R.K.: A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30(4), 921–935 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bartels, S., Carstensen, C.: A convergent adaptive finite element method for an optimal design problem. Numer. Math. 108(3), 359–385 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Carstensen, C.: Quasi-interpolation and a posteriori error analysis in finite element methods. M2AN. Math. Model. Numer. Anal. 33(6), 1187–1202 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carstensen, C., Bartels, S.: Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM. Math. Comput. 71(239), 945–969 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Carstensen, C., Gedicke, J., Mehrmann, V., Miedlar, A.: An adaptive finite element method with asymptotic saturation for eigenvalue problems. Numer. Math. 128(4), 615–634 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carstensen, C., Verfürth, R.: Edge residuals dominate a posteriori error estimates for low order finite element methods. SIAM J. Numer. Anal. 36(5), 1571–1587 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cascon, J., Kreuzer, C., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dörfler, W.: A convergent adaptive algorithm for Poisson‘s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dörfler, W., Nochetto, R.H.: Small data oscillation implies the saturation assumption. Numer. Math. 91(1), 1–12 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ferraz-Leite, S., Ortner, C., Praetorius, D.: Convergence of simple adaptive Galerkin schemes based on \(h-h/2\) error estimators. Numer. Math. 116(2), 291–316 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Rodríguez, R.: A posteriori error analysis in the finite element method. In: Finite element methods (Jyväskylä, 1993), Lecture Notes in Pure and Appl. Math., vol. 164, pp. 389–397. Dekker, New York (1994)Google Scholar
  13. 13.
    Schatz, A.H., Wang, J.P.: Some new error estimates for Ritz–Galerkin methods with minimal regularity assumptions. Math. Comput. 65(213), 19–27 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Stevenson, R.: The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77(261), 227–241 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Advances in numerical mathematics. Wiley, London (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Institut für Numerische SimulationUniversität BonnBonnGermany
  3. 3.Ruprecht-Karls-Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen (IWR)Mathematische Methoden der SimulationHeidelbergGermany

Personalised recommendations