In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of a quasilinear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of two quasilinear elliptic PDEs which have to be fulfilled at almost all times coupled with a nonsmooth, semilinear ODE that has to hold true in almost all points in space. The system is discretized by a constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. Numerical experiments are added to illustrate the proven rates of convergence.
Keywords: Error estimates, finite elements, quasilinear coupled PDE-ODE system, damage material model
@article{M2AN_2021__55_4_1347_0,
author = {Holtmannsp\"otter, Marita},
title = {\protect\emph{A priori} error estimates for the space-time finite element approximation of a quasilinear gradient enhanced damage model},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1347--1374},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {4},
doi = {10.1051/m2an/2021021},
mrnumber = {4281737},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021021/}
}
TY - JOUR AU - Holtmannspötter, Marita TI - A priori error estimates for the space-time finite element approximation of a quasilinear gradient enhanced damage model JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 1347 EP - 1374 VL - 55 IS - 4 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021021/ DO - 10.1051/m2an/2021021 LA - en ID - M2AN_2021__55_4_1347_0 ER -
%0 Journal Article %A Holtmannspötter, Marita %T A priori error estimates for the space-time finite element approximation of a quasilinear gradient enhanced damage model %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 1347-1374 %V 55 %N 4 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021021/ %R 10.1051/m2an/2021021 %G en %F M2AN_2021__55_4_1347_0
Holtmannspötter, Marita. A priori error estimates for the space-time finite element approximation of a quasilinear gradient enhanced damage model. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 4, pp. 1347-1374. doi: 10.1051/m2an/2021021
[1] , and , Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000) 797–826. | MR | Zbl | DOI
[2] and , The Mathematical Theory of Finite Element Methods. Springer Verlag (2002). | MR | Zbl | DOI
[3] , and , An adaptive finite element approximation of a generalized Ambrosio-Tortorelli functional. Math. Models Methods Appl. Sci. 23 (2013) 1663–1697. | MR | Zbl | DOI
[4] , and , Quasistatic crack growth in nonlinear elasticity. Arch. Rational Mech. Anal. 176 (2005) 165–225. | MR | Zbl | DOI
[5] and , A method for gradient enhancement of continuum damage models. Technische Mechanik, Ruhr-Universität Bochum 28 (2008) 43–52.
[6] and , A regularization framework for damage-plasticity models via gradient enhancement of the free energy. Int. J. Numer. Methods Biomed. Eng. 27 (2011) 1199–1210. | MR | Zbl | DOI
[7] and , On the rate-independent limit of systems with dry friction and small viscosity. J. Convex Anal. 13 (2006) 151–167. | MR | Zbl
[8] , Gewöhnliche und Operatordifferentialgleichungen. Vieweg, Berlin (2004).
[9] and , Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal. 28 (1991) 43–77. | MR | Zbl | DOI
[10] and , Adaptive finite element methods for parabolic problems II: optimal error estimates in and . SIAM J. Numer. Anal. 32 (1995) 706–740. | MR | Zbl | DOI
[11] and , Adaptive finite element methods for parabolic problems IV: Nonlinear problems. SIAM J. Numer. Anal. 32 (1995) 1729–1749. | MR | Zbl | DOI
[12] , and , Time discretization of parabolic problems by the discontinuous Galerkin method. Rairo M.M.a.N 19 (1985) 611–643. | MR | Zbl | Numdam
[13] and , The discontinuous Galerkin method for semilinear parabolic problems. ESAIM: M2AN 27 (1993) 35–54. | MR | Zbl | Numdam | DOI
[14] and , Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998) 1319–1342. | MR | Zbl | DOI
[15] and , Existence and convergence for quasi-static evolution in brittle fracture. Comm. Pure Appl. Math. 56 (2003) 1465–1500. | MR | Zbl | DOI
[16] , The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. (1921).
[17] , A -estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283 (1989) 679–687. | MR | Zbl | DOI
[18] , and , Higher regularity for solutions to elliptic systems in divergence form subject to mixed boundary conditions. Annali di Matematica Pura ed Applicata (2018). | MR
[19] and , A priori error estimates for the space-time finite element approximation of a non-smooth optimal control problem governed by a coupled semilinear PDE-ODE system. Preprint; arXiv: 2004.05837 (2020). | MR
[20] , and , A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Preprint; arXiv:2004.04448 (2019). | MR
[21] , , and , A fast and robust numerical treatment of a gradient-enhanced model for brittle damage. Int. J. Multiscale Comput. Eng. 17 (2019) 151–180. | DOI
[22] , and , A vanishing viscosity approach to a rate-independent damage model. Math. Models and Methods Appl. Sci. 23 (2013) 565–616. | MR | Zbl | DOI
[23] , and , The fundamental role on nonlocal and local balance laws of material forces in finite elastoplasticity and damage mechanics. Int. J. Solids Struct (2006). | Zbl
[24] and , A priori error estimates for space-time finite element discretization of parabolic optimal control problems part I: Problems without control constraints. SIAM J. Control Optim. 47 (2008) 1150–1177. | MR | Zbl | DOI
[25] and , Analysis of a viscous two-field gradient damage model, part I: Existence and uniqueness. ZAA 38 (2019) 249–286. | MR
[26] and , Analysis of a viscous two-field gradient damage model, part II: Penalization limit. ZAA 38 (2019) 439–474. | MR
[27] and , A priori error estimates for a linearized fracture control problem (2018). DOI: 10.1007/s11081-020-09574-z.
[28] , Elastoplastic-damage modelling including the gradient of damage: formulation and computational aspects. Int. J. Solids Struct. 38 (2001) 5421–5451. | Zbl | DOI
[29] and , A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numerische Mathematik 120 (2011) 345–386. | MR | Zbl | DOI
[30] , Enhanced damage modeling for fracture and fatigue. Ph.D. thesis, Technische Universiteit Eindhoven (1999).
[31] and , Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. de Gruyter (1996). | MR | Zbl | DOI
[32] , Analysis and optimal control of a damage model with penalty. Ph.D. thesis, Technische Universität Dortmund (2017).
[33] , Optimal control of a viscous two-field gradient damage model. GAMM-Mitteilungen 2018 40 (2018) 287–311. | MR | DOI
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