A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems

Du, Qiang; Ju, Lili; Lu, Jianfang

doi:10.1090/mcom/3333

A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems
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by Qiang Du, Lili Ju and Jianfang Lu;

Math. Comp. 88 (2019), 123-147

DOI: https://doi.org/10.1090/mcom/3333

Published electronically: April 12, 2018

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Abstract:

Given that nonlocal diffusion (ND) problems allow their solutions to have spatial discontinuities, discontinuous Galerkin (DG) discretizations in space become natural choices when numerical approximations are considered. In this paper, we design and study a novel DG method for solving the one-dimensional time-dependent ND problem. The key idea of the method is the introduction of an auxiliary variable, analogous to the classic local discontinuous Galerkin (LDG) method but with some nonlocal extensions. Theoretical analysis shows that the proposed semi-discrete DG scheme is $L^2$-stable, convergent, and in particular asymptotically compatible. This latter feature implies that the classical DG approximation of the local diffusion problem can be recovered in the zero horizon limit. We also present various numerical tests to demonstrate the effectiveness and robustness of our method.

References

E. Askari, F. Bobaru, R. B. Lehoucq, M. L. Parks, S. A. Silling, and O. Weckner, Peridynamics for multiscale materials modeling, Journal of Physics, Conference Series, 125 (2008), 012078.
Fuensanta Andreu-Vaillo, José M. Mazón, Julio D. Rossi, and J. Julián Toledo-Melero, Nonlocal diffusion problems, Mathematical Surveys and Monographs, vol. 165, American Mathematical Society, Providence, RI; Real Sociedad Matemática Española, Madrid, 2010. MR 2722295, DOI 10.1090/surv/165
Peter W. Bates and Adam Chmaj, An integrodifferential model for phase transitions: stationary solutions in higher space dimensions, J. Statist. Phys. 95 (1999), no. 5-6, 1119–1139. MR 1712445, DOI 10.1023/A:1004514803625
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
Bernardo Cockburn, Suchung Hou, and Chi-Wang Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case, Math. Comp. 54 (1990), no. 190, 545–581. MR 1010597, DOI 10.1090/S0025-5718-1990-1010597-0
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 520174
Bernardo Cockburn, San Yih Lin, and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems, J. Comput. Phys. 84 (1989), no. 1, 90–113. MR 1015355, DOI 10.1016/0021-9991(89)90183-6
Bernardo Cockburn and Chi-Wang Shu, The Runge-Kutta local projection $P^1$-discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 3, 337–361 (English, with French summary). MR 1103092, DOI 10.1051/m2an/1991250303371
Bernardo Cockburn and Chi-Wang Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp. 52 (1989), no. 186, 411–435. MR 983311, DOI 10.1090/S0025-5718-1989-0983311-4
Bernardo Cockburn and Chi-Wang Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems, J. Comput. Phys. 141 (1998), no. 2, 199–224. MR 1619652, DOI 10.1006/jcph.1998.5892
Bernardo Cockburn and Chi-Wang Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998), no. 6, 2440–2463. MR 1655854, DOI 10.1137/S0036142997316712
Qiang Du, Max Gunzburger, R. B. Lehoucq, and Kun Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev. 54 (2012), no. 4, 667–696. MR 3023366, DOI 10.1137/110833294
Qiang Du, Max Gunzburger, R. B. Lehoucq, and Kun Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci. 23 (2013), no. 3, 493–540. MR 3010838, DOI 10.1142/S0218202512500546
Qiang Du, Zhan Huang, and Richard B. Lehoucq, Nonlocal convection-diffusion volume-constrained problems and jump processes, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 4, 961–977. MR 3206434, DOI 10.3934/dcdsb.2014.19.961
Qiang Du and Jiang Yang, Asymptotically compatible Fourier spectral approximations of nonlocal Allen-Cahn equations, SIAM J. Numer. Anal. 54 (2016), no. 3, 1899–1919. MR 3514714, DOI 10.1137/15M1039857
Qiang Du, Jiang Yang, and Zhi Zhou, Analysis of a nonlocal-in-time parabolic equation, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), no. 2, 339–368. MR 3639119, DOI 10.3934/dcdsb.2017016
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
Paul Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in nonlinear analysis, Springer, Berlin, 2003, pp. 153–191. MR 1999098
Guy Gilboa and Stanley Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul. 6 (2007), no. 2, 595–630. MR 2338496, DOI 10.1137/060669358
Guy Gilboa and Stanley Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul. 7 (2008), no. 3, 1005–1028. MR 2480109, DOI 10.1137/070698592
Tadele Mengesha and Qiang Du, The bond-based peridynamic system with Dirichlet-type volume constraint, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 1, 161–186. MR 3164542, DOI 10.1017/S0308210512001436
W. H. Reed and T. R. Hill, Triangular mesh methods for the neutron transport equation, Tech. report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.
B. Ren, C. T. Wu, and E. Askari, A 3D discontinuous Galerkin finite element method with the bond-based peridynamics model for dynamic brittle failure analysis, International Journal of Impact Engineering 99 (2017), 14–25.
Lorenzo Rosasco, Mikhail Belkin, and Ernesto De Vito, On learning with integral operators, J. Mach. Learn. Res. 11 (2010), 905–934. MR 2600634
S. A. Silling, M. Epton, O. Weckner, J. Xu, and E. Askari, Peridynamic states and constitutive modeling, J. Elasticity 88 (2007), no. 2, 151–184. MR 2348150, DOI 10.1007/s10659-007-9125-1
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, Linearized theory of peridynamic states, J. Elasticity 99 (2010), no. 1, 85–111. MR 2592410, DOI 10.1007/s10659-009-9234-0
S. A. Silling and R. B. Lehoucq, Peridynamic theory of solid mechanics, Advances in Applied Mechanics 44 (2010), 73-168.
S. A. Silling, O. Weckner, E. Askari, and F. Bobaru, Crack nucleation in a peridynamic solid, International Journal of Fracture 162 (2010), 219–227.
S. A. Silling, M. Zimmermann, and R. Abeyaratne, Deformation of a peridynamic bar, J. Elasticity 73 (2003), no. 1-3, 173–190 (2004). MR 2057742, DOI 10.1023/B:ELAS.0000029931.03844.4f
Xiaochuan Tian and Qiang Du, Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations, SIAM J. Numer. Anal. 51 (2013), no. 6, 3458–3482. Author name corrected by publisher. MR 3143839, DOI 10.1137/13091631X
Xiaochuan Tian and Qiang Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models, SIAM J. Numer. Anal. 52 (2014), no. 4, 1641–1665. MR 3231986, DOI 10.1137/130942644
Xiaochuan Tian and Qiang Du, Nonconforming discontinuous Galerkin methods for nonlocal variational problems, SIAM J. Numer. Anal. 53 (2015), no. 2, 762–781. MR 3323545, DOI 10.1137/140978831
Xiaochuan Tian, Qiang Du, and Max Gunzburger, Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains, Adv. Comput. Math. 42 (2016), no. 6, 1363–1380. MR 3571209, DOI 10.1007/s10444-016-9466-z
Hao Tian, Lili Ju, and Qiang Du, Nonlocal convection-diffusion problems and finite element approximations, Comput. Methods Appl. Mech. Engrg. 289 (2015), 60–78. MR 3327145, DOI 10.1016/j.cma.2015.02.008
Hao Tian, Lili Ju, and Qiang Du, A conservative nonlocal convection-diffusion model and asymptotically compatible finite difference discretization, Comput. Methods Appl. Mech. Engrg. 320 (2017), 46–67. MR 3646345, DOI 10.1016/j.cma.2017.03.020
Yunzhe Tao, Xiaochuan Tian, and Qiang Du, Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations, Appl. Math. Comput. 305 (2017), 282–298. MR 3621707, DOI 10.1016/j.amc.2017.01.061
Yanxiang Zhao, Jiakou Wang, Yanping Ma, and Qiang Du, Generalized local and nonlocal master equations for some stochastic processes, Comput. Math. Appl. 71 (2016), no. 11, 2497–2512. MR 3501334, DOI 10.1016/j.camwa.2015.09.030