Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

doi:10.1007/s42967-019-00051-8

Communications on Applied Mathematics and Computation ›› 2020, Vol. 2 ›› Issue (2): 261-303.doi: 10.1007/s42967-019-00051-8

• ORIGINAL PAPER • 上一篇下一篇

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

Burak Aksoylu^1,2, Fatih Celiker², George A. Gazonas¹

1 CCDC Army Research Laboratory, Attn: FCDD-RLW-MB, Aberdeen Proving Ground, Aberdeen, MD 21005, USA;
2 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

收稿日期:2019-05-22 修回日期:2019-10-01 出版日期:2020-06-20 发布日期:2020-02-19
通讯作者: Burak Aksoylu, Fatih Celiker, George A. Gazonas E-mail:burak@wayne.edu;celiker@wayne.edu;george.a.gazonas.civ@mail.mil

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

Burak Aksoylu^1,2, Fatih Celiker², George A. Gazonas¹

1 CCDC Army Research Laboratory, Attn: FCDD-RLW-MB, Aberdeen Proving Ground, Aberdeen, MD 21005, USA;
2 Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

Received:2019-05-22 Revised:2019-10-01 Online:2020-06-20 Published:2020-02-19
Contact: Burak Aksoylu, Fatih Celiker, George A. Gazonas E-mail:burak@wayne.edu;celiker@wayne.edu;george.a.gazonas.civ@mail.mil

摘要/Abstract

摘要： We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics, a nonlocal formulation of continuum mechanics. We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation. A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero. We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible. We fnd that, for polynomial degrees greater than or equal to two, there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible. We verify these fndings through extensive numerical experiments.

关键词: Nonlocal operator, Inhomogeneous local boundary condition, Nonlocal difusion, Asymptotic compatibility, Collocation method, Peridynamics, Functional calculus

Abstract: We study the convergence and asymptotic compatibility of higher order collocation methods for nonlocal operators inspired by peridynamics, a nonlocal formulation of continuum mechanics. We prove that the methods are optimally convergent with respect to the polynomial degree of the approximation. A numerical method is said to be asymptotically compatible if the sequence of approximate solutions of the nonlocal problem converges to the solution of the corresponding local problem as the horizon and the grid sizes simultaneously approach zero. We carry out a calibration process via Taylor series expansions and a scaling of the nonlocal operator via a strain energy density argument to ensure that the resulting collocation methods are asymptotically compatible. We fnd that, for polynomial degrees greater than or equal to two, there exists a calibration constant independent of the horizon size and the grid size such that the resulting collocation methods for the nonlocal difusion are asymptotically compatible. We verify these fndings through extensive numerical experiments.

Key words: Nonlocal operator, Inhomogeneous local boundary condition, Nonlocal difusion, Asymptotic compatibility, Collocation method, Peridynamics, Functional calculus

中图分类号:

Burak Aksoylu, Fatih Celiker, George A. Gazonas. Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility[J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 261-303.

参考文献

1. Aksoylu, B., Gazonas, G.A.:Inhomogeneous local boundary conditions in nonlocal problems. In:Proceedings of ECCOMAS2018, 6th European Conference on Computational Mechanics (ECCM 6) and 7th European Conference on Computational Fluid Dynamics (ECFD 7), 11-15 June 2018, Glasgow, UK (In press)
2. Aksoylu, B., Gazonas, G.A.:On nonlocal problems with inhomogeneous local boundary conditions. J. Peridyn. Nonlocal Model (In press)
3. Aksoylu, B., Celiker, F.:Comparison of nonlocal operators utilizing perturbation analysis. In:Karasozen, B., Manguogiu, M., Tezer-Sezgin, M., Goktepe, S. (eds.) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol. 112, pp. 589-606. Springer, Berlin (2016). https://doi.org/10.1007/978-3-319-39929-4_57
4. Aksoylu, B., Celiker, F.:Nonlocal problems with local Dirichlet and Neumann boundary conditions. J. Mech. Mater. Struct. 12(4), 425-437 (2017). https://doi.org/10.2140/jomms.2017.12.425
5. Aksoylu, B., Kaya, A.:Conditioning and error analysis of nonlocal problems with local boundary conditions. J. Comput. Appl. Math. 335, 1-19 (2018). https://doi.org/10.1016/j.cam.2017.11.023
6. Aksoylu, B., Unlu, Z.:Conditioning analysis of nonlocal integral operators in fractional Sobolev spaces. SIAM J. Numer. Anal. 52(2), 653-677 (2014). https://doi.org/10.1137/13092407X
7. Aksoylu, B., Beyer, H.R., Celiker, F.:Application and implementation of incorporating local boundary conditions into nonlocal problems. Numer. Funct. Anal. Optim. 38(9), 1077-1114 (2017). https://doi. org/10.1080/01630563.2017.1320674
8. Aksoylu, B., Beyer, H.R., Celiker, F.:Theoretical foundations of incorporating local boundary conditions into nonlocal problems. Rep. Math. Phys. 40(1), 39-71 (2017). https://doi.org/10.1016/S0034-4877(17)30061-7
9. Aksoylu, B., Celiker, F., Kilicer, O.:Nonlocal operators with local boundary conditions:an overview. In:Voyiadjis, G. (eds.) Handbook of Nonlocal Continuum Mechanics for Materials and Structures, pp. 1293-1330. Springer, Cham (2019). https://doi.org/10.1007/978-3-319-58729-5_34
10. Aksoylu, B., Celiker, F., Kilicer, O.:Nonlocal problems with local boundary conditions in higher dimensions. Adv. Comp. Math. 45(1), 453-492 (2019). https://doi.org/10.1007/s10444-018-9624-6
11. Atkinson, K., Han, W.:Theoretical Numerical Analysis. Springer, New York (2009)
12. Beyer, H.R., Aksoylu, B., Celiker, F.:On a class of nonlocal wave equations from applications. J. Math. Phys. 57(6), 062902 (2016). https://doi.org/10.1063/1.4953252
13. Cortazar, C., Elgueta, M., Rossi, J.D., Wolanski, N.:How to approximate the heat equation with Neumann boundary conditions by nonlocal difusion problems. Arch. Rational Mech. Anal. 187(1), 137-156 (2008). https://doi.org/10.1007/s00205-007-0062-8
14. D'Elia, M., Tian, X., Yu, Y.:A physically-consistent, fexible and efcient strategy to convert local boundary conditions into nonlocal volume constraints (2019). arXiv:1906.04259 (Preprint)
15. Du, Q., Yang, J.:Asymptotically compatible fourier spectral approximations of nonlocal Allen-Cahn equations. SIAM J. Numer. Anal. 54(3), 1899-1919 (2016)
16. Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.:Analysis and approximation of nonlocal difusion problems with volume constraints. SIAM Rev. 54, 667-696 (2012)
17. Du, Q., Lu, X.H., Lu, J., Tian, X.:A quasinonlocal coupling method for nonlocal and local difusion models. SIAM J. Numer. Anal. 56(3), 1386-1404 (2018)
18. Du, Q., Ju, L., Lu, J.:A discontinuous Galerkin method for one-dimensional time-dependent nonlocal difusion problems. Math. Comput. 88(315), 123-147 (2019)
19. Seleson, P., Parks, M.L.:On the role of the infuence function in the peridynamic theory. Int. J. Multiscale Comput. Eng. 9(6), 689-706 (2011)
20. Silling, S.:Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175-209 (2000)
21. Silling, S.A.:Introduction to peridynamics. In:Bobaru, F., Foster, J.T., Geubelle, P.H., Silling, S.A. (eds.) Handbook of Peridynamic Modeling, Advances in Applied Mathematics, pp. 25-60. Chapman and Hall, London (2017). https://doi.org/10.1201/9781315373331
22. Tao, Y., Tian, X., Du, Q.:Nonlocal difusion and peridynamic models with Neumann type constraints and their numerical approximations. Appl. Math. Comput. 305, 282-298 (2017)
23. Tian, X., Du, Q.:Analysis and comparison of diferent approximations to nonlocal difusion and linear peridynamic equations. SIAM J. Numer. Anal. 51(6), 3458-3482 (2013)
24. Tian, X., Du, Q.:Asymptotically compatible schemes and applications to robust discretization of nonlocal problems. SIAM J. Numer. Anal. 52(4), 1641-1665 (2014)
25. Tian, H., Ju, L., Du, Q.:Nonlocal convection-difusion problems and fnite element approximations. Comput. Methods Appl. Mech. Eng. 289, 60-78 (2015)
26. Tian, X., Du, Q., Gunzburger, M.:Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal difusion problems on bounded domains. Adv. Comput. Math. 42(6), 1363-1380 (2016)
27. Tian, H., Ju, L., Du, Q.:A conservative nonlocal convection-difusion model and asymptotically compatible fnite diference discretization. Comput. Methods Appl. Mech. Eng. 320, 46-67 (2017)
28. You, H., Lu, X., Trask, N., Yu, Y.:An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems (2019). arXiv:1908.03853 (Preprint)
29. Zhang, X., Wu, J., Ju, L.:An accurate and asymptotically compatible collocation scheme for nonlocal difusion problems. Appl. Numer. Math. 133, 52-68 (2018)

[1]	Qingqing Wu, Xiaoyan Zeng. Jacobi Collocation Methods for Solving Generalized Space-Fractional Burgers' Equations[J]. Communications on Applied Mathematics and Computation, 2020, 2(2): 305-318.
[2]	Qiang Du, Lili Ju, Jianfang Lu, Xiaochuan Tian. A Discontinuous Galerkin Method with Penalty for One-Dimensional Nonlocal Difusion Problems[J]. Communications on Applied Mathematics and Computation, 2020, 2(1): 31-55.

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

Higher Order Collocation Methods for Nonlocal Problems and Their Asymptotic Compatibility

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 2

编辑推荐

Metrics

本文评价