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Energy Norm A Posteriori Error Estimates

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Error Estimates for Advanced Galerkin Methods

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 88))

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Abstract

This chapter provides deeper insights into the verification part of the computational V&V strategy introduced in Sect. 1.1. To be more precise, we investigate the question whether the boundary value problems derived in Chaps. 2 and 3 are solved right by the Galerkin methods presented in Chaps. 4 and 5, i.e. the (mixed) finite element method (based on SCNI), the extended finite element method, and the meshfree element-free Galerkin and reproducing kernel particle methods. For the time being, we restrict our considerations to the linearized elasticity problem (3.28) because this linear problem allows for the development of verification strategies in a more convenient way. Verification strategies applied to the finite hyperelasticity problem will be detailed in Chap. 8.

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Notes

  1. 1.

    For the sake of brevity, the subscripts “low”, “upp”, and “app” are frequently suppressed and used only whenever we need to distinguish between lower and upper error bounds and error approximations.

References

  • Ainsworth, M., Oden, J.T.: A unified approach to a posteriori error estimation based on element residual methods. Numer. Math. 65, 23–50 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, New York (2000)

    Google Scholar 

  • Alonso, A.: Error estimators for a mixed method. Numer. Math. 74, 385–395 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér. 19, 7–32 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  • Babuška, I., Banerjee, U., Osborn, J.E., Li, Q.: Quadrature for meshless methods. Int. J. Numer. Meth. Engng. 76, 1434–1470 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Babuška, I., Banerjee, U., Osborn, J.E., Zhang, Q.: Effect of numerical integration on meshless methods. Comput. Methods Appl. Mech. Engrg. 198, 2886–2897 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Babuška, I., Miller, A.: A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator. Comput. Methods Appl. Mech. Engrg. 61, 1–40 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • Babuška, I., Rheinboldt, W.C.: A-posteriori error estimates for the finite element method. Int. J. Numer. Meth. Engng. 12, 1597–1615 (1978a)

    Article  MATH  Google Scholar 

  • Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15, 736–754 (1978b)

    Article  MathSciNet  MATH  Google Scholar 

  • Babuška, I., Rheinboldt, W.C.: Analysis of optimal finite-element meshes in \(R^1\). Math. Comp. 33, 435–463 (1979)

    MathSciNet  MATH  Google Scholar 

  • Babuška, I., Strouboulis, T.: The Finite Element Method and its Reliability. Oxford University Press, Oxford (2001)

    Google Scholar 

  • Babuška, I., Strouboulis, T., Upadhyay, C.S.: A model study of the quality of a posteriori error estimators for linear elliptic problems. Error estimation in the interior of patchwise uniform grids of triangles. Comput. Methods Appl. Mech. Engrg. 114, 307–378 (1994)

    Article  MathSciNet  Google Scholar 

  • Bahriawati, C., Carstensen, C.: Three MATLAB implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control. Comput. Methods Appl. Math. 5, 333–361 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2003)

    MATH  Google Scholar 

  • Bank, R.E., Smith, R.K.: A posteriori error estimates based on hierarchical bases. SIAM J. Numer. Anal. 30, 921–935 (1993)

    MathSciNet  MATH  Google Scholar 

  • Bank, R.E., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283–301 (1985)

    MathSciNet  MATH  Google Scholar 

  • Bordas, S., Duflot, M.: Derivative recovery and a posteriori error estimate for extended finite elements. Comput. Methods Appl. Mech. Engrg. 196, 3381–3399 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Braess, D., Verfürth, R.: A posteriori error estimators for the Raviart-Thomas element. SIAM J. Numer. Anal. 33, 2431–2444 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Bramble, J.H., Xu, J.: A local post-processing technique for improving the accuracy in mixed finite-element approximations. SIAM J. Numer. Anal. 26, 1267–1275 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Brauchli, H.J., Oden, J.T.: On the calculation of consistent stress distributions in finite element applications. Int. J. Numer. Meth. Engng. 3, 317–325 (1971)

    Article  MATH  Google Scholar 

  • Brezzi, F., J. Douglas Jr., J., Durán, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51, 237–250 (1987)

    Google Scholar 

  • Brink, U., Stein, E.: A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems. Comput. Methods Appl. Mech. Engrg. 161, 77–101 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Bufler, H., Stein, E.: Zur Plattenberechnung mittels finiter Elemente. Ing. Archiv 39, 248–260 (1970)

    Article  MATH  Google Scholar 

  • Cai, Z., Korsawe, J., Starke, G.: An adaptive least squares mixed finite element method for the stress-displacement formulation of linear elasticity. Numer. Methods Partial Differential Eq. 21, 132–148 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Carstensen, C.: A posteriori error estimate for the mixed finite element method. Math. Comp. 66, 465–476 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Carstensen, C., Funken, S.A.: Constants in Clément-interpolation error and residual based a posteriori error estimates in finite element methods. East-West J. Numer. Math. 8, 153–175 (2000)

    MathSciNet  MATH  Google Scholar 

  • Carstensen, C., Funken, S.A.: Averaging technique for FE - a posteriori error control in elasticity. Part I: Conforming FEM. Comput. Methods Appl. Mech. Engrg. 190, 2483–2498 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Chamoin, L., Díez, P. (eds.): Verifying Calculations—Forty Years On—An Overview of Classical Verification Techniques for FEM Simulations. Springer, New York (2016)

    Google Scholar 

  • Chen, J.S., Hillman, M., Rüter, M.: An arbitrary order variationally consistent integration for Galerkin meshfree methods. Int. J. Numer. Meth. Engng. 95, 387–418 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Chung, H.J., Belytschko, T.: An error estimate in the EFG method. Comput. Mech. 21, 91–100 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Clément, P.: Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9, 77–84 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  • Dolbow, J., Belytschko, T.: Volumetric locking in the element free Galerkin method. Int. J. Numer. Meth. Engng. 46, 925–942 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Duarte, C.A., Oden, J.T.: \(H\)-\(p\) clouds—an \(h\)-\(p\) meshless method. Numer. Methods Partial Differential Eq. 12, 673–705 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Duflot, M., Bordas, S.: A posteriori error estimation for extended finite elements by an extended global recovery. Int. J. Numer. Meth. Engng. 76, 1123–1138 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Gabaldón, F., Goicolea, J.M.: Linear and non-linear finite element error estimation based on assumed strain fields. Int. J. Numer. Meth. Engng. 55, 413–429 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Gavete, L., Cuesta, J.L., Ruiz, A.: A procedure for approximation of the error in the EFG method. Int. J. Numer. Meth. Engng. 53, 677–690 (2002)

    Google Scholar 

  • Gavete, L., Gavete, M.L., Alonso, B., Martín, A.J.: A posteriori error approximation in EFG method. Int. J. Numer. Meth. Engng. 58, 2239–2263 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Gerasimov, T., Rüter, M., Stein, E.: An explicit residual-type error estimator for \(\mathbb{Q}_1\)-quadrilateral extended finite element method in two-dimensional linear elastic fracture mechanics. Int. J. Numer. Meth. Engng. 90, 1118–1155 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Gerasimov, T., Stein, E., Wriggers, P.: Constant-free explicit error estimator with sharp upper error bound property for adaptive FE analysis in elasticity and fracture. Int. J. Numer. Meth. Engng. 101, 79–126 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  • González-Estrada, O.A., Natarajan, S., Ródenas, J.J., Nguyen-Xuan, H., Bordas, S.P.A.: Efficient recovery-based error estimation for the smoothed finite element method for smooth and singular linear elasticity. Comput. Mech. 52, 37–52 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Häussler-Combe, U., Korn, C.: An adaptive approach with the Element-Free-Galerkin method. Comput. Methods Appl. Mech. Engrg. 162, 203–222 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Hild, P., Lleras, V., Renard, Y.: A residual error estimator for the XFEM approximation of the elasticity problem. Submitted to Computational Mechanics (2011)

    Google Scholar 

  • Johnson, C., Hansbo, P.: Adaptive finite element methods in computational mechanics. Comput. Methods Appl. Mech. Engrg. 101, 143–181 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Kelly, D.W.: The self-equilibration of residuals and complementary a posteriori error estimates in the finite element method. Int. J. Numer. Meth. Engng. 20, 1491–1506 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  • Kondratiev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points (in Russian). Trudy Moskovskogo Matematicheskogo Obshchestva 16, 209–292 (1967)

    Google Scholar 

  • Krongauz, Y., Belytschko, T.: Consistent pseudo-derivatives in meshless methods. Comput. Methods Appl. Mech. Engrg. 146, 371–386 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Ladevèze, P., Leguillon, D.: Error estimate procedure in the finite element method and applications. SIAM J. Numer. Anal. 20, 485–509 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Ladevèze, P., Pelle, J.-P.: Mastering Calculations in Linear and Nonlinear Mechanics. Springer, New York (2005)

    MATH  Google Scholar 

  • Larsson, F., Hansbo, P., Runesson, K.: Strategies for computing goal-oriented a posteriori error measures in non-linear elasticity. Int. J. Numer. Meth. Engng. 55, 879–894 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Le, C.V., Askes, H., Gilbert, M.: Adaptive element-free Galerkin method applied to the limit analysis of plates. Comput. Methods Appl. Mech. Engrg. 199, 2487–2496 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Lee, C.K., Shuai, Y.Y.: An automatic adaptive refinement procedure for the reproducing kernel particle method. Part I: Stress recovery and a posteriori error estimation. Comput. Mech. 40, 399–413 (2007)

    Article  MATH  Google Scholar 

  • Liu, W.K., Li, S., Belytschko, T.: Moving least-square reproducing kernel methods (I) Methodology and convergence. Comput. Methods Appl. Mech. Engrg. 143, 113–154 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Lovadina, C., Stenberg, R.: Energy norm a posteriori error estimates for mixed finite element methods. Math. Comp. 75, 1659–1674 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Luo, Y., Häussler-Combe, U.: A gradient-based adaptation procedure and its implementation in the element-free Galerkin method. Int. J. Numer. Meth. Engng. 56, 1335–1354 (2003)

    Article  MATH  Google Scholar 

  • Neittaanmäki, P., Repin, S.: Reliable Methods for Computer Simulation—Error Control and A Posteriori Estimates. Elsevier B.V., Amsterdam (2004)

    MATH  Google Scholar 

  • Nguyen-Thoi, T., Liu, G.R., Nguyen-Xuan, H., Nguyen-Tran, C.: Adaptive analysis using the node-based smoothed finite element method (NS-FEM). Int. J. Numer. Meth. Bio. 27, 198–218 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Parés, N., Díez, P., Huerta, A.: Subdomain-based flux-free a posteriori error estimators. Comput. Methods Appl. Mech. Engrg. 195, 297–323 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Prager, W., Synge, J.L.: Approximations in elasticity based on the concept of function space. Quart. Appl. Math. 5, 241–269 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  • Prange, C., Loehnert, S., Wriggers, P.: Error estimation for crack simulations using the XFEM. Int. J. Numer. Meth. Engng. 91, 1459–1474 (2012)

    Article  MathSciNet  Google Scholar 

  • Ródenas, J.J., González-Estrada, O.A., Díez, P., Fuenmayor, F.J.: Accurate recovery-based upper error bounds for the extended finite element framework. Comput. Methods Appl. Mech. Engrg. 199, 2607–2621 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Ródenas, J.J., González-Estrada, O.A., Tarancón, J.E., Fuenmayor, F.J.: A recovery-type error estimator for the extended finite element method based on singular+smooth stress field splitting. Int. J. Numer. Meth. Engng. 76, 545–571 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Ródenas, J.J., Tur, M., Fuenmayor, F.J., Vercher, A.: Improvement of the superconvergent patch recovery technique by the use of constraint equations: The SPR-C technique. Int. J. Numer. Meth. Engng. 70, 705–727 (2007)

    Article  MATH  Google Scholar 

  • Rodríguez, R.: Some remarks on Zienkiewicz-Zhu estimator. Numer. Methods Partial Differential Eq. 10, 625–635 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Rüter, M., Hillman, M., Chen, J.S.: Corrected stabilized non-conforming nodal integration in meshfree methods. In: Griebel, M., Schweitzer, M.A. (eds.) Meshfree Methods for Partial Differential Equations VI, pp. 75–92. Springer, Berlin (2012)

    MATH  Google Scholar 

  • Rüter, M., Stenberg, R.: Error-controlled adaptive mixed finite element methods for second-order elliptic equations. Comput. Mech. 42, 447–456 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Rüter, M.O., Chen, J.S.: An enhanced-strain error estimator for Galerkin meshfree methods based on stabilized conforming nodal integration. Comput. Math. Appl. 74, 2144–2171 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Schröder, J., Schwarz, A., Steeger, K.: Least-squares mixed finite element formulations for isotropic and anisotropic elasticity at small and large strains. In: Schröder, J., Wriggers, P. (eds.) Advanced Finite Element Technologies, pp. 131–175. Springer, Berlin (2016)

    Chapter  MATH  Google Scholar 

  • Solomon, J.: Numerical Algorithms—Methods for Computer Vision, Machine Learning, and Graphics. CRC Press, Boca Raton (2015)

    Google Scholar 

  • Stein, E. (ed.): Error-controlled Adaptive Finite Elements in Solid Mechanics. John Wiley & Sons, Chichester (2003)

    Google Scholar 

  • Stein, E., Ahmad, R.: An equilibrium method for stress calculation using finite element methods in solid- and structural-mechanics. Comput. Methods Appl. Mech. Engrg. 10, 175–198 (1977)

    Article  MATH  Google Scholar 

  • Stein, E., Ohnimus, S.: Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems. Comput. Methods Appl. Mech. Engrg. 176, 363–385 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Stein, E., Rust, W.: Mesh adaptations for linear 2D finite-element discretizations in structural mechanics, especially in thin shell analysis. J. Comput. Appl. Math. 36, 107–129 (1991)

    Article  MATH  Google Scholar 

  • Stein, E., Rüter, M.: Computational mechanics with model adaptivity in analysis and design: Current research and perspectives. In: Ibrahimbegovic, A., Brank, B. (eds.) Engineering Structures under Extreme Conditions—Multi-Physics and Multi-Scale Computer Models in Non-Linear Analysis and Optimal Design, pp. 383–406. IOS Press, Amsterdam (2005)

    Google Scholar 

  • Stein, E., Rüter, M.: Finite element methods for elasticity with error-controlled discretization and model adaptivity. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 3, 2nd edn., pp. 5–100. John Wiley & Sons, Chichester (2017)

    Google Scholar 

  • Stein, E., Rüter, M., Ohnimus, S.: Adaptive finite element analysis and modelling of solids and structures. Findings, problems and trends. Int. J. Numer. Meth. Engng. 60, 103–138 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Stein, E., Rüter, M., Ohnimus, S.: Error-controlled adaptive goal-oriented modeling and finite element approximations in elasticity. Comput. Methods Appl. Mech. Engrg. 196, 3598–3613 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Stein, E., Rüter, M., Ohnimus, S.: Implicit upper bound error estimates for combined expansive model and discretization adaptivity. Comput. Methods Appl. Mech. Engrg. 200, 2626–2638 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Stenberg, R.: Some new families of finite elements for the Stokes equations. Numer. Math. 56, 827–838 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Stenberg, R.: Postprocessing schemes for some mixed finite elements. RAIRO Modél. Math. Anal. Numér. 25, 151–167 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  • Sukumar, N., Moran, B., Belytschko, T.: The natural element method in solid mechanics. Int. J. Numer. Meth. Engng. 43, 839–887 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Verfürth, R.: A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50, 67–83 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press, Oxford (2013)

    Book  MATH  Google Scholar 

  • Vidal, Y., Parés, N., Díez, P., Huerta, A.: Bounds for quantities of interest and adaptivity in the element-free Galerkin method. Int. J. Numer. Meth. Engng. 76, 1782–1818 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Article  MathSciNet  MATH  Google Scholar 

  • You, Y., Chen, J.S., Lu, H.: Filters, reproducing kernel, and adaptive meshfree method. Comput. Mech. 31, 316–326 (2003)

    Google Scholar 

  • Yvonnet, J., Coffignal, G., Ryckelynck, D., Lorong, P., Chinesta, F.: A simple error indicator for meshfree methods based on natural neighbors. Comput. & Struct. 84, 1301–1312 (2006)

    Article  MathSciNet  Google Scholar 

  • Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Meth. Engng. 24, 337–357 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int. J. Numer. Meth. Engng. 33, 1331–1364 (1992a)

    Article  MathSciNet  MATH  Google Scholar 

  • Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity. Int. J. Numer. Meth. Engng. 33, 1365–1382 (1992b)

    Article  MathSciNet  MATH  Google Scholar 

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    Marcus Olavi Rüter

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Rüter, M.O. (2019). Energy Norm A Posteriori Error Estimates. In: Error Estimates for Advanced Galerkin Methods. Lecture Notes in Applied and Computational Mechanics, vol 88. Springer, Cham. https://doi.org/10.1007/978-3-030-06173-9_6

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