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Introduction

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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

In this introductory chapter, the topics of this monograph are briefly discussed and embedded into the bigger picture of computational validation and verification strategies in Computational Mechanics. More precisely, different types of errors are introduced that appear during the numerical simulation process of a physical phenomenon based on various Galerkin methods. The Galerkin methods dealt with in this monograph are the (conventional) finite element method (FEM) and, in particular, advanced versions of the finite element method, such as the extended finite element method (XFEM).

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References

  • Ainsworth, M.: A posteriori error estimation for fully discrete hierarchic models of elliptic boundary value problems on thin domains. Numer. Math. 80, 325–362 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • 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)

    Book  MATH  Google Scholar 

  • Atluri, S.N., Zhu, T.: A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22, 117–127 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Aubin, J.-P.: Behavior of the error of the approximate solution of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods. Ann. Scuola Norm. Sup. Pisa 21, 599–637 (1967)

    MATH  Google Scholar 

  • Auricchio, F., Beirão da Veiga, L., Brezzi, F., Lovadina, C.: Mixed finite element methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 1, 2nd edn., pp. 149–201. John Wiley & Sons, Chichester (2017)

    Google Scholar 

  • Babuška, I., Melenk, J.M.: The partition of unity method. Int. J. Numer. Meth. Engng. 40, 727–758 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Babuška, I., Miller, A.: The post-processing approach in the finite element method. Part 3: A posteriori error estimates and adaptive mesh selection. Int. J. Numer. Meth. Engng. 20, 2311–2324 (1984)

    Article  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., Schwab, C.: A posteriori error estimation for hierarchic models of elliptic boundary value problems on thin domains. SIAM J. Numer. Anal. 33, 221–246 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Book  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)

    Article  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)

    Article  MathSciNet  MATH  Google Scholar 

  • Barsoum, R.S.: On the use of isoparametric finite elements in linear fracture mechanics. Int. J. Numer. Meth. Engng. 10, 25–37 (1976)

    Article  MATH  Google Scholar 

  • Becker, R., Rannacher, R.: A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West J. Numer. Math. 4, 237–264 (1996)

    MathSciNet  MATH  Google Scholar 

  • Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Meth. Engng. 45, 601–620 (1999)

    Article  MATH  Google Scholar 

  • Belytschko, T., Lu, Y.Y., Gu, L.: Element-free Galerkin methods. Int. J. Numer. Meth. Engng. 37, 229–256 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    Book  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., Hu, H.Y., Chi, S.W.: The role of quadrature in meshfree methods: Variational consistency in Galerkin weak form and collocation in strong form. IACM Expr. 34, 11–16 (2013)

    Google Scholar 

  • Chen, J.S., Liu, W.K., Hillman, M.C., Chi, S.W., Lian, Y., Bessa, M.A.: Reproducing kernel particle method for solving partial differential equations. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 2, 2nd edn., pp. 691–734. John Wiley & Sons, Chichester (2017)

    Google Scholar 

  • Chen, J.S., Pan, C., Wu, C.T., Liu, W.K.: Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures. Comput. Methods Appl. Mech. Engrg. 139, 195–227 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Chen, J.S., Wu, C.T., Yoon, S., You, Y.: A stabilized conforming nodal integration for Galerkin mesh-free methods. Int. J. Numer. Meth. Engng. 50, 435–466 (2001)

    Article  MATH  Google Scholar 

  • Cherepanov, G.P.: Crack propagation in continuous media. J. Appl. Math. Mech. 31, 503–512 (1967)

    Article  MATH  Google Scholar 

  • Cirak, F., Ramm, E.: A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem. Comput. Methods Appl. Mech. Engrg. 156, 351–362 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Courant, R.: Variational methods for the solution of problems of equilibrium and vibrations. Bull. Amer. Math. Soc. 49, 1–23 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  • Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Introduction to adaptive methods for differential equations. Acta Numer. 106–158 (1995)

    Google Scholar 

  • Eshelby, J.D.: The force on an elastic singularity. Philos. Trans. Roy. Soc. London Ser. A 244, 87–112 (1951)

    Google Scholar 

  • Galerkin, B.G.: Series solutions of some cases of equilibrium of elastic beams and plates (in Russian). Vestn. Inshenernov. 1, 897–908 (1915)

    Google Scholar 

  • Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. astr. Soc. 181, 375–389 (1977)

    Article  MATH  Google Scholar 

  • Heintz, P., Larsson, F., Hansbo, P., Runesson, K.: Adaptive strategies and error control for computing material forces in fracture mechanics. Int. J. Numer. Meth. Engng. 60, 1287–1299 (2004)

    Article  MATH  Google Scholar 

  • Henshell, R.D., Shaw, K.G.: Crack tip finite elements are unnecessary. Int. J. Numer. Meth. Engng. 9, 495–507 (1975)

    Article  MATH  Google Scholar 

  • Huerta, A., Belytschko, T., Fernández-Méndez, S., Rabczuk, T., Zhuang, X., Arroyo, M.: Meshfree methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 2, 2nd edn., pp. 653–690. John Wiley & Sons, Chichester (2017)

    Google Scholar 

  • Koenke, C., Harte, R., Krätzig, W.B., Rosenstein, O.: On adaptive remeshing techniques for crack simulation problems. Engrg. Comput. 15, 74–88 (1998)

    Article  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 

  • Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comp. 37, 141–158 (1981)

    Article  MathSciNet  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 

  • Larsson, F., Runesson, K.: Modeling and discretization errors in hyperelasto-(visco-)plasticity with a view to hierarchical modeling. Comput. Methods Appl. Mech. Engrg. 193, 5283–5300 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, G.R., Dai, K.Y., Nguyen, T.T.: A smoothed finite element method for mechanics problems. Comput. Mech. 39, 859–877 (2007)

    Article  MATH  Google Scholar 

  • Liu, W.K., Jun, S., Li, S., Adee, J., Belytschko, T.: Reproducing kernel particle methods for structural dynamics. Int. J. Numer. Meth. Engng. 38, 1655–1679 (1995a)

    Article  MathSciNet  MATH  Google Scholar 

  • Liu, W.K., Jun, S., Zhang, Y.F.: Reproducing kernel particle methods. Int. J. Numer. Meth. Fluids 20, 1081–1106 (1995b)

    Article  MathSciNet  MATH  Google Scholar 

  • Lu, Y.Y., Belytschko, T., Gu, L.: A new implementation of the element free Galerkin method. Comput. Methods Appl. Mech. Engrg. 113, 397–414 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Lucy, L.B.: A numerical approach to the testing of the fission hypothesis. Astron. J. 82, 1013–1024 (1977)

    Article  Google Scholar 

  • Maugin, G.A.: Material Inhomogeneities in Elasticity. Chapman & Hall, London (1993)

    Book  MATH  Google Scholar 

  • Melenk, J.M., Babuška, I.: The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Engrg. 139, 289–314 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Engng. 46, 131–150 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Moës, N., Dolbow, J.E., Sukumar, N.: Extended finite element methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 3, 2nd edn., pp 173–193, John Wiley & Sons, Chichester (2017)

    Google Scholar 

  • Murthy, K.S.R.K., Mukhopadhyay, M.: Adaptive finite element analysis of mixed-mode crack problems with automatic mesh generator. Int. J. Numer. Meth. Engng. 49, 1087–1100 (2000)

    Article  MATH  Google Scholar 

  • Nayroles, B., Touzot, G., Villon, P.: Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10, 307–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    Chapter  MATH  Google Scholar 

  • Nitsche, J.A.: Ein Kriterium für die Quasioptimalität des Ritzschen Verfahrens. Numer. Math. 11, 346–348 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  • Oden, J.T., Prudhomme, S.: Estimation of modeling error in computational mechanics. J. Comput. Phys. 182, 496–515 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Oden, J.T., Vemaganti, K.: Estimation of local modeling error and goal-oriented modeling of heterogeneous materials; Part I: error estimates and adaptive algorithms. J. Comput. Phys. 164, 22–47 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Ohnimus, S., Stein, E., Walhorn, E.: Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems. Int. J. Numer. Meth. Engng. 52, 727–746 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Organ, D., Fleming, M., Terry, T., Belytschko, T.: Continuous meshless approximations for nonconvex bodies by diffraction and transparency. Comput. Mech. 18, 225–235 (1996)

    Article  MATH  Google Scholar 

  • Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2002)

    MATH  Google Scholar 

  • Osher, S., Sethian, J.A.: Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Paraschivoiu, M., Patera, A.T.: A hierarchical duality approach to bounds for the outputs of partial differential equations. Comput. Methods Appl. Mech. Engrg. 158, 389–407 (1998)

    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 

  • Prasad, M.V.K.V., Krishnamoorthy, C.S.: Adaptive finite element analysis of mode I fracture in cement-based materials. Int. J. Numer. Anal. Meth. Geomech. 25, 1131–1147 (2001)

    Article  MATH  Google Scholar 

  • Prudhomme, S., Oden, J.T.: On goal-oriented error estimation for local elliptic problems: application to the control of pointwise errors. Comput. Methods Appl. Mech. Engrg. 176, 313–331 (1999)

    Google Scholar 

  • Randles, P.W., Libersky, L.D.: Smoothed Particle Hydrodynamics: Some recent improvements and applications. Comput. Methods Appl. Mech. Engrg. 139, 375–408 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  • Rannacher, R., Suttmeier, F.-T.: A posteriori error estimation and mesh adaptation for finite element models in elasto-plasticity. Comput. Methods Appl. Mech. Engrg. 176, 333–361 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Rice, J.R.: A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379–386 (1968)

    Article  Google Scholar 

  • Roache, P.J.: Quantification of uncertainty in computational fluid dynamics. Annu. Rev. Fluid. Mech. 29, 123–160 (1997)

    Article  MathSciNet  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., Stein, E.: Goal-oriented a posteriori error estimates in linear elastic fracture mechanics. Comput. Methods Appl. Mech. Engrg. 195, 251–278 (2006)

    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 

  • Simo, J.C., Rifai, M.S.: A class of assumed strain methods and the method of incompatible modes. Int. J. Numer. Meth. Engng. 29, 1595–1638 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  • Stein, E., Ohnimus, S.: Coupled model- and solution-adaptivity in the finite-element method. Comput. Methods Appl. Mech. Engrg. 150, 327–350 (1997)

    Article  MathSciNet  MATH  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 

  • Steinmann, P.: Application of material forces to hyperelastostatic fracture mechanics. I. Continuum mechanical setting. Int. J. Solids Structures 37, 7371–7391 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Stern, M., Becker, E.B.: A conforming crack tip element with quadratic variation in the singular fields. Int. J. Numer. Meth. Engng. 12, 279–288 (1978)

    Article  MATH  Google Scholar 

  • Stone, T.J., Babuška, I.: A numerical method with a posteriori error estimation for determining the path taken by a propagating crack. Comput. Methods Appl. Mech. Engrg. 160, 245–271 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Strouboulis, T., Babuška, I., Copps, K.: The design and analysis of the Generalized Finite Element Method. Comput. Methods Appl. Mech. Engrg. 181, 43–69 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Strouboulis, T., Copps, K., Babuška, I.: The generalized finite element method. Comput. Methods Appl. Mech. Engrg. 190, 4081–4193 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Turner, M.J., Clough, R.W., Martin, H.C., Topp, L.J.: Stiffness and deflection analysis of complex structures. Journal Aero. Sci. 23, 805–823 (1956)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  • Wiberg, N.E., Abdulwahab, F.: Patch recovery based on superconvergent derivatives and equilibrium. Int. J. Numer. Meth. Engng. 36, 2703–2724 (1993)

    Article  MathSciNet  MATH  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 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Civil and Environmental Engineering, University of California, Los Angeles, USA

    Marcus Olavi Rüter

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Rüter, M.O. (2019). Introduction. 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_1

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