Stability and convergence of time-stepping methods for a nonlocal model for diffusion

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July 2015, 20(5): 1315-1335. doi: 10.3934/dcdsb.2015.20.1315

Stability and convergence of time-stepping methods for a nonlocal model for diffusion

Qingguang Guan ^1, and Max Gunzburger ^1,

1.	Department of Scientific Computing, Florida State University, Tallahassee, FL 32306, United States, United States

Received November 2013 Revised January 2015 Published May 2015

Abstract
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A time-dependent nonlocal model for diffusion is considered. A feature of the model is that instead of boundary conditions, constraints over regions having finite measures are imposed. The explicit forward-Euler, implicit backward-Euler, and Crank-Nicolson methods are considered for discretizing the time derivative and piecewise-linear finite element methods are used for spatial discretization. The unconditional stability of the backward-Euler and Crank-Nicolson schemes and the conditional stability of the forward-Euler scheme are proved as are optimal error estimates for all three schemes. Comparisons with the analogous results for classical local diffusion problems, e.g., the heat equation, are provided as are the results of numerical experiments that illustrate the theoretical results.

Keywords: stability, Nonlocal diffusion, error estimates., superdiffusion, convergence, finite element methods.

Mathematics Subject Classification: Primary: 65M60, 76R50; Secondary: 45K0.

Citation: Qingguang Guan, Max Gunzburger. Stability and convergence of time-stepping methods for a nonlocal model for diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1315-1335. doi: 10.3934/dcdsb.2015.20.1315

References:

[1]	B. Aksoylu and T. Mengesha, Results on nonlocal boundary value problems, Numer. Funct. Anal. Optim., 31 (2010), 1301-1317. doi: 10.1080/01630563.2010.519136.
[2]	A. Buades, B. Coll and J. Morel, Image denoising methods: A new nonlocal principle, SIAM Review, 52 (2010), 113-147. doi: 10.1137/090773908.
[3]	X. Chen and M. Gunzburger, Continuous and discontinuous finite element methods for a peridynamics model of mechanics, Comput. Meth. Appl. Mech. Engrg., 200 (2011), 1237-1250. doi: 10.1016/j.cma.2010.10.014.
[4]	Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54 (2012), 667-696. doi: 10.1137/110833294.
[5]	Q. Du, L. Tian and X. Zhao, A Convergent Adaptive Finite Element Algorithm for Nonlocal Diffusion and Peridynamic Models, SIAM J. Numer. Anal., 51 (2013), 1211-1234. doi: 10.1137/120871638.
[6]	V. Ervin, N. Heuer and J. Roop, Numerical approximation of a time dependent, non-linear, fractional order diffusion equation, SIAM J. Math. Anal., 45 (2007), 572-591. doi: 10.1137/050642757.
[7]	G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630. doi: 10.1137/060669358.
[8]	G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592.
[9]	M. Gunzburger and R. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598. doi: 10.1137/090766607.
[10]	Y. Lou, X. Zhang, S. Osher and A. Bertozzi, Image recovery via nonlocal operators, J. Sci. Comput., 42 (2010), 185-197. doi: 10.1007/s10915-009-9320-2.
[11]	H. Wang and H. Tian, A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model, J. Comput. Phys., 231 (2012), 7730-7738. doi: 10.1016/j.jcp.2012.06.009.
[12]	O. Weckner and R. Abeyaratne, The effect of long-range forces on the dynamics of a bar, J. Mech. Phys. Solids., 53 (2005), 705-728. doi: 10.1016/j.jmps.2004.08.006.
[13]	K. Zhou and Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions, SIAM J. Math. Anal., 48 (2010), 1759-1780. doi: 10.1137/090781267.
[14]	K. Zhou and Q. Du, Mathematical analysis for the peridynamic nonlocal continuum theory, Math. Model. Numer. Anal., 45 (2011), 217-234. doi: 10.1051/m2an/2010040.

show all references

References:

[1]	B. Aksoylu and T. Mengesha, Results on nonlocal boundary value problems, Numer. Funct. Anal. Optim., 31 (2010), 1301-1317. doi: 10.1080/01630563.2010.519136.
[2]	A. Buades, B. Coll and J. Morel, Image denoising methods: A new nonlocal principle, SIAM Review, 52 (2010), 113-147. doi: 10.1137/090773908.
[3]	X. Chen and M. Gunzburger, Continuous and discontinuous finite element methods for a peridynamics model of mechanics, Comput. Meth. Appl. Mech. Engrg., 200 (2011), 1237-1250. doi: 10.1016/j.cma.2010.10.014.
[4]	Q. Du, M. Gunzburger, R. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Review, 54 (2012), 667-696. doi: 10.1137/110833294.
[5]	Q. Du, L. Tian and X. Zhao, A Convergent Adaptive Finite Element Algorithm for Nonlocal Diffusion and Peridynamic Models, SIAM J. Numer. Anal., 51 (2013), 1211-1234. doi: 10.1137/120871638.
[6]	V. Ervin, N. Heuer and J. Roop, Numerical approximation of a time dependent, non-linear, fractional order diffusion equation, SIAM J. Math. Anal., 45 (2007), 572-591. doi: 10.1137/050642757.
[7]	G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, Multiscale Model. Simul., 6 (2007), 595-630. doi: 10.1137/060669358.
[8]	G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028. doi: 10.1137/070698592.
[9]	M. Gunzburger and R. Lehoucq, A nonlocal vector calculus with application to nonlocal boundary value problems, Multiscale Model. Simul., 8 (2010), 1581-1598. doi: 10.1137/090766607.
[10]	Y. Lou, X. Zhang, S. Osher and A. Bertozzi, Image recovery via nonlocal operators, J. Sci. Comput., 42 (2010), 185-197. doi: 10.1007/s10915-009-9320-2.
[11]	H. Wang and H. Tian, A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model, J. Comput. Phys., 231 (2012), 7730-7738. doi: 10.1016/j.jcp.2012.06.009.
[12]	O. Weckner and R. Abeyaratne, The effect of long-range forces on the dynamics of a bar, J. Mech. Phys. Solids., 53 (2005), 705-728. doi: 10.1016/j.jmps.2004.08.006.
[13]	K. Zhou and Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions, SIAM J. Math. Anal., 48 (2010), 1759-1780. doi: 10.1137/090781267.
[14]	K. Zhou and Q. Du, Mathematical analysis for the peridynamic nonlocal continuum theory, Math. Model. Numer. Anal., 45 (2011), 217-234. doi: 10.1051/m2an/2010040.

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