A posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models

Du, Qiang; Ju, Lili; Tian, Li; Zhou, Kun

doi:10.1090/S0025-5718-2013-02708-1

A posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models
HTML articles powered by AMS MathViewer

by Qiang Du, Lili Ju, Li Tian and Kun Zhou PDF

Math. Comp. 82 (2013), 1889-1922 Request permission

Abstract:

In this paper, we present some results on a posteriori error analysis of finite element methods for solving linear nonlocal diffusion and bond-based peridynamic models. In particular, we aim to propose a general abstract frame work for a posteriori error analysis of the peridynamic problems. A posteriori error estimators are consequently prompted, the reliability and efficiency of the estimators are proved. Connections between nonlocal a posteriori error estimation and classical local estimation are studied within continuous finite element space. Numerical experiments (1D) are also given to test the theoretical conclusions.

References

Mark Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000. MR 1885308, DOI 10.1002/9781118032824
Burak Aksoylu and Michael L. Parks, Variational theory and domain decomposition for nonlocal problems, Appl. Math. Comput. 217 (2011), no. 14, 6498–6515. MR 2773238, DOI 10.1016/j.amc.2011.01.027
Burak Aksoylu and Tadele Mengesha, Results on nonlocal boundary value problems, Numer. Funct. Anal. Optim. 31 (2010), no. 12, 1301–1317. MR 2738853, DOI 10.1080/01630563.2010.519136
Bacim Alali and Robert Lipton, Multiscale dynamics of heterogeneous media in the peridynamic formulation, J. Elasticity 106 (2012), no. 1, 71–103. MR 2870298, DOI 10.1007/s10659-010-9291-4
F. Andreu, J. M. Mazón, J. D. Rossi, and J. Toledo, A nonlocal $p$-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions, SIAM J. Math. Anal. 40 (2008/09), no. 5, 1815–1851. MR 2471902, DOI 10.1137/080720991
E. Askari, F. Bobaru, R. B. Lehoucq, M. L. Parks, S. A. Silling, and O. Weckner, Peridynamics for multiscale materials modeling, J. Phys. : Conf. Ser. 125 (2008), DOI:10. 1088/1742-6596/125/1/012078.
Gilles Aubert and Pierre Kornprobst, Can the nonlocal characterization of Sobolev spaces by Bourgain et al. be useful for solving variational problems?, SIAM J. Numer. Anal. 47 (2009), no. 2, 844–860. MR 2485435, DOI 10.1137/070696751
I. Babuška and A. Miller, A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator, Comput. Methods Appl. Mech. Engrg. 61 (1987), no. 1, 1–40. MR 880421, DOI 10.1016/0045-7825(87)90114-9
I. Babuska and W. C. Rheinboldt, A posteriori error estimates for the finite element method, Internat. J. Numer. Methods Engrg. 12 (1978), 1597–1615.
Ivo Babuška and Werner C. Rheinboldt, A posteriori error analysis of finite element solutions for one-dimensional problems, SIAM J. Numer. Anal. 18 (1981), no. 3, 565–589. MR 615532, DOI 10.1137/0718036
I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978), no. 4, 736–754. MR 483395, DOI 10.1137/0715049
S. Bartels, C. Carstensen, and G. Dolzmann, Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis, Numer. Math. 99 (2004), no. 1, 1–24. MR 2101782, DOI 10.1007/s00211-004-0548-3
F. Bobaru, L. A. Alves, S. A. Silling, and A. Askari, Dynamic crack branching and adaptive refinement in peridynamics, 8th World Congress on Computational Mechanics, 5th European Congress on Computational Methods in Applied Sciences and Engineering, June 30-July 5, 2008, Venice, Italy.
F. Bobaru, M. Yang, L. F. Alves, S. A. Silling, E. Askari, and J. Xu, Convergence, adaptive refinement, and scaling in 1d peridynamics, Intern. J. Numer. Meth. Engrg. 77 (2009), 852–877.
J. Bourgain, H. Brezis, and P. Mironescu, Another look at Sobolev spaces, in: J. L. Menaldi, E. Rofman, A. Sulem (Eds. ), Optimal Control and Partial Differential Equations. A Volume in Honour of A. Bensoussan’s 60th Birthday, IOS Press (2001), 439-455.
N. Burch and R. Lehoucq, Classical, nonlocal, and fractional diffusion equations on bounded domains, Sandia Technical Report, SAND 2010-1762J, Sandia National Laboratories, 2010.
Carsten Carstensen, Efficiency of a posteriori BEM-error estimates for first-kind integral equations on quasi-uniform meshes, Math. Comp. 65 (1996), no. 213, 69–84. MR 1320892, DOI 10.1090/S0025-5718-96-00671-0
C. Carstensen, All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable, Math. Comp. 73 (2004), no. 247, 1153–1165. MR 2047082, DOI 10.1090/S0025-5718-03-01580-1
C. Carstensen and B. P. Rynne, A posteriori error control for finite element approximations of the integral equation for thin wire antennas, ZAMM Z. Angew. Math. Mech. 82 (2002), no. 4, 284–288. MR 1900489, DOI 10.1002/1521-4001(200204)82:4<284::AID-ZAMM284>3.0.CO;2-5
X. Chen and Max Gunzburger, Continuous and discontinuous finite element methods for a peridynamics model of mechanics, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 9-12, 1237–1250. MR 2796157, DOI 10.1016/j.cma.2010.10.014
Alan Demlow, Local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems, Math. Comp. 76 (2007), no. 257, 19–42. MR 2261010, DOI 10.1090/S0025-5718-06-01879-5
Q. Du, M. Gunzburger, R. B. Lehoucq, and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and the nonlocal balance laws, Math. Mod. Meth. Appl. Sci. 23 (2013), pp. 493–540.
Q. Du, M. Gunzburger, R. B. Lehoucq, and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review 56 (2012), pp. 676–696.
Q. Du, M. Gunzburger, R. B. Lehoucq, and K. Zhou, Analysis of the volume-constrained peridynamic Navier equation of linear elasticity, Journal of Elasticity, 2013 (online), doi:10.1007/s10659-012-9418-x.
Qiang Du and Kun Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory, ESAIM Math. Model. Numer. Anal. 45 (2011), no. 2, 217–234. MR 2804637, DOI 10.1051/m2an/2010040
Etienne Emmrich and Olaf Weckner, Analysis and numerical approximation of an integro-differential equation modeling non-local effects in linear elasticity, Math. Mech. Solids 12 (2007), no. 4, 363–384. MR 2349159, DOI 10.1177/1081286505059748
E. Emmrich and O. Weckner, The peridynamic equation of motion in nonlocal elasticity theory, III European Conference on Computational Mechanics - Solids, Structures and Coupled Problems in Engineering, 2006.
Etienne Emmrich and Olaf Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity, Commun. Math. Sci. 5 (2007), no. 4, 851–864. MR 2375050
Max Gunzburger and R. B. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul. 8 (2010), no. 5, 1581–1598. MR 2728700, DOI 10.1137/090766607
Lili Ju, Max Gunzburger, and Weidong Zhao, Adaptive finite element methods for elliptic PDEs based on conforming centroidal Voronoi-Delaunay triangulations, SIAM J. Sci. Comput. 28 (2006), no. 6, 2023–2053. MR 2272250, DOI 10.1137/050643568
R. B. Lehoucq and S. A. Silling, Force flux and the peridynamic stress tensor, J. Mech. Phys. Solids 56 (2008), no. 4, 1566–1577. MR 2404022, DOI 10.1016/j.jmps.2007.08.004
Richard W. Macek and Stewart A. Silling, Peridynamics via finite element analysis, Finite Elem. Anal. Des. 43 (2007), no. 15, 1169–1178. MR 2393568, DOI 10.1016/j.finel.2007.08.012
Ricardo H. Nochetto, Tobias von Petersdorff, and Chen-Song Zhang, A posteriori error analysis for a class of integral equations and variational inequalities, Numer. Math. 116 (2010), no. 3, 519–552. MR 2684296, DOI 10.1007/s00211-010-0310-y
S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids 48 (2000), no. 1, 175–209. MR 1727557, DOI 10.1016/S0022-5096(99)00029-0
S. A. Silling, A coarsening method for linear peridynamics, International Journal for Multiscale Computational Engineering 9 (2011), pp. 609–622.
S. A. Silling and E. Askari, A meshfree method based on the peridynamic model of solid mechanics, Computers & Structures 83 (2005), Advances in Meshfree Methods, 1526–1535.
S. A. Silling and R. B. Lehoucq, Convergence of peridynamics to classical elasticity theory, J. Elasticity 93 (2008), no. 1, 13–37. MR 2430855, DOI 10.1007/s10659-008-9163-3
R. Verfürth, a review of a posteriori error estimation and adaptive mesh-refinement techniques, Wiley-Teubner, Chichester, UK, 1996.
K. Yu, X. J. Xin, and K. B. Lease, A new method of adaptive integration with error control for bond-based peridynamics, Proceedings of the World Congress on Engineering and Computer Science 2010, Vol II, San Francisco, USA.
Kun Zhou and Qiang Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions, SIAM J. Numer. Anal. 48 (2010), no. 5, 1759–1780. MR 2733097, DOI 10.1137/090781267