Estimating and localizing the algebraic and total numerical errors using flux reconstructions

Article

Abstract

This paper presents a methodology for computing upper and lower bounds for both the algebraic and total errors in the context of the conforming finite element discretization of the Poisson model problem and an arbitrary iterative algebraic solver. The derived bounds do not contain any unspecified constants and allow estimating the local distribution of both errors over the computational domain. Combining these bounds, we also obtain guaranteed upper and lower bounds on the discretization error. This allows to propose novel mathematically justified stopping criteria for iterative algebraic solvers ensuring that the algebraic error will lie below the discretization one. Our upper algebraic and total error bounds are based on locally reconstructed fluxes in \({\mathbf {H}}(\mathrm{div},\varOmega )\), whereas the lower algebraic and total error bounds rely on locally constructed \(H^1_0(\varOmega )\)-liftings of the algebraic and total residuals. We prove global and local efficiency of the upper bound on the total error and its robustness with respect to the approximation polynomial degree. Relationships to the previously published estimates on the algebraic error are discussed. Theoretical results are illustrated on numerical experiments for higher-order finite element approximations and the preconditioned conjugate gradient method. They in particular witness that the proposed methodology yields a tight estimate on the local distribution of the algebraic and total errors over the computational domain and illustrate the associated cost.

Keywords

Numerical solution of partial differential equations Finite element method A posteriori error estimation Algebraic error Discretization error Stopping criteria Spatial distribution of the error 

Mathematics Subject Classification

65N15 65N30 76M10 65N22 65F10 

References

  1. 1.
    Ainsworth, M.: Robust a posteriori error estimation for nonconforming finite element approximation. SIAM J. Numer. Anal. 42, 2320–2341 (2005)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Arioli, M., Georgoulis, E.H., Loghin, D.: Stopping criteria for adaptive finite element solvers. SIAM J. Sci. Comput. 35, A1537–A1559 (2013)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Arioli, M., Liesen, J., Miȩdlar, A., Strakoš, Z.: Interplay between discretization and algebraic computation in adaptive numerical solution of elliptic PDE problems. GAMM Mitt. 36, 102–129 (2013)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Babuška, I., Strouboulis, T.: The Finite Element Method and Its Reliability, Numerical Mathematics and Scientific Computation. The Clarendon Press, New York (2001)MATHGoogle Scholar
  5. 5.
    Becker, R., Johnson, C., Rannacher, R.: Adaptive error control for multigrid finite element methods. Computing 55, 271–288 (1995)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Becker, R., Mao, S.: Convergence and quasi-optimal complexity of a simple adaptive finite element method, M2AN Math. Model. Numer. Anal. 43, 1203–1219 (2009)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Becker, R., Mao, S., Shi, Z.: A convergent nonconforming adaptive finite element method with quasi-optimal complexity. SIAM J. Numer. Anal. 47, 4639–4659 (2010)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Berndt, M., Manteuffel, T.A., McCormick, S.F.: Local error estimates and adaptive refinement for first-order system least squares (FOSLS). Electron. Trans. Numer. Anal. 6, 35–43 (1997). Special issue on multilevel methods (Copper Mountain, CO, 1997)MATHMathSciNetGoogle Scholar
  9. 9.
    Braess, D., Pillwein, V., Schöberl, J.: Equilibrated residual error estimates are \(p\)-robust. Comput. Methods Appl. Mech. Engrg. 198, 1189–1197 (2009)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Braess, D., Schöberl, J.: Equilibrated residual error estimator for edge elements. Math. Comp. 77, 651–672 (2008)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Burstedde, C., Kunoth, A.: A wavelet-based nested iteration-inexact conjugate gradient algorithm for adaptively solving elliptic PDEs. Numer. Algorithms 48, 161–188 (2008)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Calvetti, D., Morigi, S., Reichel, L., Sgallari, F.: Computable error bounds and estimates for the conjugate gradient method. Numer. Algorithms 25, 75–88 (2000)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Cancès, C., Pop, I.S., Vohralík, M.: An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow. Math. Comp. 83, 153–188 (2014)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Carstensen, C., Funken, S.A.: Fully reliable localized error control in the FEM. SIAM J. Sci. Comput. 21, 1465–1484 (2000)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Ciarlet, P.G.: The finite element method for elliptic problems. In: Classics in Applied Mathematics, vol. 40 . Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). Reprint of the 1978 original (North-Holland, Amsterdam)Google Scholar
  16. 16.
    Ciarlet, P.G.: Linear and Nonlinear Functional Analysis with Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA (2013)MATHGoogle Scholar
  17. 17.
    Dahlquist, G., Golub, G.H., Nash, S.G.: Bounds for the Error in Linear Systems. In: Semi-infinite Programming (Proceedings of Workshop, Bad Honnef 1978) Lecture Notes in Control and Information Science, vol. 15, pp. 154–172 . Springer, Berlin (1979)Google Scholar
  18. 18.
    Destuynder, P., Métivet, B.: Explicit error bounds in a conforming finite element method. Math. Comp. 68, 1379–1396 (1999)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Deuflhard, P.: Cascadic conjugate gradient methods for elliptic partial differential equations: algorithm and numerical results. In: Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993) vol. 180 of Contemp. Math, pp. 29–42. American Mathematical Society, Providence, RI (1994)Google Scholar
  20. 20.
    Dolean, V., Jolivet, P., Nataf, F.: An Introduction to Domain Decomposition Methods: Algorithms, Theory, and Parallel Implementation. Other Titles in Applied Mathematics. SIAM, Philadelphia (2015)CrossRefMATHGoogle Scholar
  21. 21.
    Dolejší, V., Šebestová, I., Vohralík, M.: Algebraic and discretization error estimation by equilibrated fluxes for discontinuous Galerkin methods on nonmatching grids. J. Sci. Comput. 64, 1–34 (2015)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Dolejší, V., Ern, A., Vohralík, M.: \(hp\)-adaptation driven by polynomial-degree-robust a posteriori error estimates for elliptic problems. SIAM J. Sci. Comput. 38, A3220–A3246 (2016)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Ern, A., Vohralík, M.: Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAM J. Sci. Comput. 35, A1761–A1791 (2013)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Ern, A., Vohralík, M.: Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53, 1058–1081 (2015)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Ern, A., Vohralík, M.: Stable broken \(H^1\) and \({\varvec {H}}({\rm div})\) polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions. HAL Preprint 01422204, submitted for publication (2016)Google Scholar
  27. 27.
    Gergelits, T., Strakoš, Z.: Composite convergence bounds based on Chebyshev polynomials and finite precision conjugate gradient computations. Numer. Algorithms 65, 759–782 (2014)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Golub, G.H., Meurant, G.: Matrices, moments and quadrature. In: Numerical Analysis 1993 (Dundee, 1993), vol. 303 of Pitman Research Notes in Mathematics Series, pp. 105–156. Longman Sci. Tech., Harlow (1994)Google Scholar
  29. 29.
    Golub, G.H., Meurant, G.: Matrices, moments and quadrature. II. How to compute the norm of the error in iterative methods. BIT 37, 687–705 (1997)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Golub, G.H., Strakoš, Z.: Estimates in quadratic formulas. Numer. Algorithms 8, 241–268 (1994)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Harbrecht, H., Schneider, R.: On error estimation in finite element methods without having Galerkin orthogonality, Berichtsreihe des SFB 611 457, Universität Bonn (2009)Google Scholar
  32. 32.
    Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. Res. Nat. Bur. Stand. 49, 409–436 (1952)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Hiptmair, R.: Operator preconditioning. Comput. Math. Appl. 52, 699–706 (2006)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Jiránek, P., Strakoš, Z., Vohralík, M.: A posteriori error estimates including algebraic error and stopping criteria for iterative solvers. SIAM J. Sci. Comput. 32, 1567–1590 (2010)CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Kellogg, R.B.: On the Poisson equation with intersecting interfaces. Appl. Anal. 4, 101–129 (1974/75) (collection of articles dedicated to Nikolai Ivanovich Muskhelishvili)Google Scholar
  36. 36.
    Liesen, J., Strakoš, Z.: Krylov Subspace Methods: Principles and Analysis, Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)MATHGoogle Scholar
  37. 37.
    Luce, R., Wohlmuth, B.I.: A local a posteriori error estimator based on equilibrated fluxes. SIAM J. Numer. Anal. 42, 1394–1414 (2004)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Málek, J., Strakoš, Z.: Preconditioning and the conjugate gradient method in the context of solving PDEs, vol. 1 of SIAM Spotlights. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2015)MATHGoogle Scholar
  39. 39.
    Meurant, G.: The computation of bounds for the norm of the error in the conjugate gradient algorithm. Numer. Algorithms 16, 77–87 (1998)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Meurant, G.: Numerical experiments in computing bounds for the norm of the error in the preconditioned conjugate gradient algorithm. Numer. Algorithms 22, 353–365 (1999)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Meurant, G., Strakoš, Z.: The Lanczos and conjugate gradient algorithms in finite precision arithmetic. Acta Numer. 15, 471–542 (2006)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Meurant, G., Tichý, P.: On computing quadrature-based bounds for the \(A\)-norm of the error in conjugate gradients. Numer. Algorithms 62, 163–191 (2013)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Morin, P., Nochetto, R.H., Siebert, K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44, 631–658 (2003). Revised reprint of “Data oscillation and convergence of adaptive FEM” [SIAM J. Numer. Anal. 38 (2000), no. 2, 466–488]CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Papež, J.: Algebraic Error in Matrix Computations in the Context of Numerical Solution of Partial Differential Equations. Ph.D. thesis, Charles University, Prague, Nov (2016)Google Scholar
  45. 45.
    Papež, J., Strakoš, Z.: On a residual-based a posteriori error estimator for the total error. IMA J. Numer. Anal. (2017) (accepted for publication)Google Scholar
  46. 46.
    Papež, J., Liesen, J., Strakoš, Z.: Distribution of the discretization and algebraic error in numerical solution of partial differential equations. Linear Algebra Appl. 449, 89–114 (2014)CrossRefMATHMathSciNetGoogle Scholar
  47. 47.
    Papež, J., Rüde, U., Vohralík, M., Wohlmuth, B.: Sharp algebraic and total a posteriori error bounds for hp nite elements via a multilevel approach. (2017) (in preparation)Google Scholar
  48. 48.
    Patera, A.T., Rønquist, E.M.: A general output bound result: application to discretization and iteration error estimation and control. Math. Models Methods Appl. Sci. 11, 685–712 (2001)CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    Payne, L.E., Weinberger, H.F.: An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5, 286–292 (1960)CrossRefMATHGoogle Scholar
  50. 50.
    Rannacher, R.: Error control in finite element computations. An introduction to error estimation and mesh-size adaptation. In: Error control and adaptivity in scientic computing (Antalya, 1998), vol. 536 of NATO Science Series C: Mathematical and Physical Sciences, pp. 247–278. Kluwer Acad. Publ., Dordrecht (1999)Google Scholar
  51. 51.
    Rektorys, K.: Variational Methods in Mathematics, Science and Engineering, 2nd edn. D. Reidel Publishing Co., Dordrecht-Boston, Mass (1980). (translated from the Czech by Michael Basch)MATHGoogle Scholar
  52. 52.
    Repin, S.: A Posteriori Estimates for Partial Differential Equations. Radon Series on Computational and Applied Mathematics, vol. 4. Walter de Gruyter GmbH & Co. KG, Berlin (2008)CrossRefGoogle Scholar
  53. 53.
    Shaidurov, V.V.: Some estimates of the rate of convergence for the cascadic conjugate-gradient method. Comput. Math. Appl. 31, 161–171 (1996)CrossRefMATHMathSciNetGoogle Scholar
  54. 54.
    Silvester, D.J., Simoncini, V.: An optimal iterative solver for symmetric indefinite systems stemming from mixed approximation. ACM Trans. Math. Softw. 37, Art. 42, 22 (2011)CrossRefMATHMathSciNetGoogle Scholar
  55. 55.
    Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7, 245–269 (2007)CrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    Strakoš, Z., Tichý, P.: On error estimation in the conjugate gradient method and why it works in finite precision computations. Electron. Trans. Numer. Anal. 13, 56–80 (2002)MATHMathSciNetGoogle Scholar
  57. 57.
    Strakoš, Z., Tichý, P.: Error estimation in preconditioned conjugate gradients. BIT 45, 789–817 (2005)CrossRefMATHMathSciNetGoogle Scholar
  58. 58.
    Veeser, A., Verfürth, R.: Poincaré constants for finite element stars. IMA J. Numer. Anal. 32, 30–47 (2012)CrossRefMATHMathSciNetGoogle Scholar
  59. 59.
    Verfürth, R.: A posteriori error estimation techniques for finite element methods. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013)CrossRefMATHGoogle Scholar
  60. 60.
    Vohralík, M., Wheeler, M.F.: A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows. Comput. Geosci. 17, 789–812 (2013)CrossRefMathSciNetGoogle Scholar
  61. 61.
    Wohlmuth, B.I., Hoppe, R.H.W.: A comparison of a posteriori error estimators for mixed finite element discretizations by Raviart-Thomas elements. Math. Comp. 68, 1347–1378 (1999)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPragueCzech Republic
  2. 2.Institute of Computer ScienceCzech Academy of SciencesPragueCzech Republic
  3. 3.Inria Paris, 2 rue Simone IffParisFrance
  4. 4.Université Paris-Est, CERMICS (ENPC)Marne-la-Vallée 2France

Personalised recommendations