Abstract
We study a natural alternating method of Schwarz type (domain decomposition) for a certain class of couplings between local and nonlocal operators. We show that our method fits into Lions’s framework and prove, as a consequence, convergence in both the continuous and the discrete settings.
Dedicated to Thomas Apel on the occasion of his 60th birthday.
Funding source: Agencia Nacional de Promoción de la Investigación, el Desarrollo Tecnológico y la Innovación
Award Identifier / Grant number: PICT 2018 – 3017
Funding source: Ministerio de Ciencia, Tecnología e Innovación Productiva
Award Identifier / Grant number: 22MATH-04
Funding source: Fondo Nacional de Desarrollo Científico y Tecnológico
Award Identifier / Grant number: 3220254
Funding source: Consejo Nacional de Investigaciones Científicas y Técnicas
Award Identifier / Grant number: 11220150100036CO
Funding source: Secretaria de Ciencia y Tecnica, Universidad de Buenos Aires
Award Identifier / Grant number: 20020160100155BA
Funding statement: G. Acosta was partially supported by ANPCyT under grant PICT 2018 – 3017 (Argentina) and MATHAMSUD 22MATH-04. F. M. Bersetche was partially supported by ANID through FONDECYT Project 3220254, by ANPCyT under grant PICT 2018 – 3017 (Argentina) and MATHAMSUD 22MATH-04. J. D. Rossi was partially supported by CONICET grant PIP GI No. 11220150100036CO (Argentina), PICT 2018 – 3183 (Argentina), UBACyT grant 20020160100155BA (Argentina) and MATHAMSUD 22MATH-04.
References
[1] G. Acosta, F. Bersetche and J. D. Rossi, Local and nonlocal energy-based coupling models, SIAM J. Math. Anal. 54 (2022), no. 6, 6288–6322. 10.1137/21M1431977Search in Google Scholar
[2] G. Acosta, F. M. Bersetche and J. P. Borthagaray, A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian, Comput. Math. Appl. 74 (2017), no. 4, 784–816. 10.1016/j.camwa.2017.05.026Search in Google Scholar
[3] F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Math. Surveys Monogr. 165, American Mathematical Society, Providence, 2010. 10.1090/surv/165Search in Google Scholar
[4] Y. Azdoud, F. Han and G. Lubineau, A morphing framework to couple non-local and local anisotropic continua, Int. J. Solids Structures 50 (2013), no. 9, 1332–1341. 10.1016/j.ijsolstr.2013.01.016Search in Google Scholar
[5] S. Badia, P. Bochev, R. Lehoucq, M. Parks, J. Fish, M. A. Nuggehally and M. Gunzburger, A forcebased blending model for atomistic-to-continuum coupling, Multiscale Model. Simul. 5 (2007), no. 5, 387–406. 10.1615/IntJMultCompEng.v5.i5.30Search in Google Scholar
[6] S. Badia, M. Parks, P. Bochev, M. Gunzburger and R. Lehoucq, On atomistic-to-continuum coupling by blending, Multiscale Model. Simul. 7 (2008), no. 1, 381–406. 10.1137/07069969XSearch in Google Scholar
[7] P. W. Bates and A. Chmaj, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions, J. Stat. Phys. 95 (1999), no. 5–6, 1119–1139. 10.1023/A:1004514803625Search in Google Scholar
[8] H. Berestycki, A.-C. Coulon, J.-M. Roquejoffre and L. Rossi, The effect of a line with nonlocal diffusion on Fisher-KPP propagation, Math. Models Methods Appl. Sci. 25 (2015), no. 13, 2519–2562. 10.1142/S0218202515400175Search in Google Scholar
[9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2010. 10.1007/978-0-387-70914-7Search in Google Scholar
[10] C. Carrillo and P. Fife, Spatial effects in discrete generation population models, J. Math. Biol. 50 (2005), no. 2, 161–188. 10.1007/s00285-004-0284-4Search in Google Scholar PubMed
[11] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9) 86 (2006), no. 3, 271–291. 10.1016/j.matpur.2006.04.005Search in Google Scholar
[12] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, Boundary fluxes for nonlocal diffusion, J. Differential Equations 234 (2007), no. 2, 360–390. 10.1016/j.jde.2006.12.002Search in Google Scholar
[13] C. Cortazar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Arch. Ration. Mech. Anal. 187 (2008), no. 1, 137–156. 10.1007/s00205-007-0062-8Search in Google Scholar
[14] M. D’Elia and P. Bochev, Formulation, analysis and computation of an optimization-based local-to-nonlocal coupling method, preprint (2019), https://arxiv.org/abs/1910.11214. 10.2172/1572228Search in Google Scholar
[15] M. D’Elia, X. Li, P. Seleson, X. Tian and Y. Yu, A review of local-to-nonlocal coupling methods in nonlocal diffusion and nonlocal mechanics, J. Peridyn. Nonlocal Model. 4 (2022), no. 1, 1–50. 10.1007/s42102-020-00038-7Search in Google Scholar
[16] M. D’Elia, M. Perego, P. Bochev and D. Littlewood, A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions, Comput. Math. Appl. 71 (2016), no. 11, 2218–2230. 10.1016/j.camwa.2015.12.006Search in Google Scholar
[17] M. D’Elia, D. Ridzal, K. J. Peterson, P. Bochev and M. Shashkov, Optimization-based mesh correction with volume and convexity constraints, J. Comput. Phys. 313 (2016), 455–477. 10.1016/j.jcp.2016.02.050Search in Google Scholar
[18] M. Di Paola, G. Failla and M. Zingales, Physically-based approach to the mechanics of strong non-local linear elasticity theory, J. Elasticity 97 (2009), no. 2, 103–130. 10.1007/s10659-009-9211-7Search in Google Scholar
[19] Q. Du, X. H. Li, J. Lu and X. Tian, A quasi-nonlocal coupling method for nonlocal and local diffusion models, SIAM J. Numer. Anal. 56 (2018), no. 3, 1386–1404. 10.1137/17M1124012Search in Google Scholar
[20] L. C. Evans, Partial Differential Equations, 2nd ed., Grad. Stud. Math. 19, American Mathematical Society, Providence, 2010. Search in Google Scholar
[21] P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, erlin (2003), 153–191. 10.1007/978-3-662-05281-5_3Search in Google Scholar
[22] C. G. Gal and M. Warma, Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces, Comm. Partial Differential Equations 42 (2017), no. 4, 579–625. 10.1080/03605302.2017.1295060Search in Google Scholar
[23] M. J. Gander, Schwarz methods over the course of time, Electron. Trans. Numer. Anal. 31 (2008), 228–255. Search in Google Scholar
[24] A. Gárriz, F. Quirós and J. D. Rossi, Coupling local and nonlocal evolution equations, Calc. Var. Partial Differential Equations 59 (2020), no. 4, Paper No. 112. 10.1007/s00526-020-01771-zSearch in Google Scholar
[25] F. Han and G. Lubineau, Coupling of nonlocal and local continuum models by the Arlequin approach, Internat. J. Numer. Methods Engrg. 89 (2012), no. 6, 671–685. 10.1002/nme.3255Search in Google Scholar
[26] V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol. 47 (2003), no. 6, 483–517. 10.1007/s00285-003-0210-1Search in Google Scholar PubMed
[27] D. Kriventsov, Regularity for a local-nonlocal transmission problem, Arch. Ration. Mech. Anal. 217 (2015), no. 3, 1103–1195. 10.1007/s00205-015-0851-4Search in Google Scholar
[28] P.-L. Lions, On the Schwarz alternating method. I, First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia (1988), 1–42. Search in Google Scholar
[29] T. Mengesha and Q. Du, The bond-based peridynamic system with Dirichlet-type volume constraint, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 1, 161–186. 10.1017/S0308210512001436Search in Google Scholar
[30] H. A. Schwarz, Über einen Grenzübergang durch alternierendes Verfahren, Vierteljahrsschrift Naturforschenden Ges. Zürich 15 (1870), 272–286. Search in Google Scholar
[31] P. Seleson and M. Gunzburger, Bridging methods for atomistic-to-continuum coupling and their implementation, Commun. Comput. Phys. 7 (2010), no. 4, 831–876. 10.4208/cicp.2009.09.053Search in Google Scholar
[32] P. Seleson, M. Gunzburger and M. L. Parks, Interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains, Comput. Methods Appl. Mech. Engrg. 266 (2013), 185–204. 10.1016/j.cma.2013.05.018Search in Google Scholar
[33] S. A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces, J. Mech. Phys. Solids 48 (2000), no. 1, 175–209. 10.1016/S0022-5096(99)00029-0Search in Google Scholar
[34] 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. 10.1007/s10659-007-9125-1Search in Google Scholar
[35] S. A. Silling and R. B. Lehoucq, Peridynamic theory of solid mechanics, Adv. Appl. Mech. 44 (2010), 73–168. 10.1016/S0065-2156(10)44002-8Search in Google Scholar
[36] C. Strickland, G. Dangelmayr and P. D. Shipman, Modeling the presence probability of invasive plant species with nonlocal dispersal, J. Math. Biol. 69 (2014), no. 2, 267–294. 10.1007/s00285-013-0693-3Search in Google Scholar PubMed
[37] X. Wang, Metastability and stability of patterns in a convolution model for phase transitions, J. Differential Equations 183 (2002), no. 2, 434–461. 10.1006/jdeq.2001.4129Search in Google Scholar
[38] L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks, J. Differential Equations 197 (2004), no. 1, 162–196. 10.1016/S0022-0396(03)00170-0Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Recent Advances in Finite Element Methods
- A Domain Decomposition Scheme for Couplings Between Local and Nonlocal Equations
- Novel Raviart–Thomas Basis Functions on Anisotropic Finite Elements
- A Cost-Efficient Space-Time Adaptive Algorithm for Coupled Flow and Transport
- Error Estimates for the Numerical Approximation of Unregularized Sparse Parabolic Control Problems
- An Analysis of High-Frequency Helmholtz Problems in Domains with Conical Points and Their Finite Element Discretisation
- Implicit Runge–Kutta Schemes for Optimal Control Problems with Evolution Equations
- A Multilevel Extension of the GDSW Overlapping Schwarz Preconditioner in Two Dimensions
- A Numerical Assessment of Finite Element Discretizations for Convection-Diffusion-Reaction Equations Satisfying Discrete Maximum Principles
- Robust Finite Element Discretization and Solvers for Distributed Elliptic Optimal Control Problems
- Finite Element Approximations for PDEs with Irregular Dirichlet Boundary Data on Boundary Concentrated Meshes
Articles in the same Issue
- Frontmatter
- Recent Advances in Finite Element Methods
- A Domain Decomposition Scheme for Couplings Between Local and Nonlocal Equations
- Novel Raviart–Thomas Basis Functions on Anisotropic Finite Elements
- A Cost-Efficient Space-Time Adaptive Algorithm for Coupled Flow and Transport
- Error Estimates for the Numerical Approximation of Unregularized Sparse Parabolic Control Problems
- An Analysis of High-Frequency Helmholtz Problems in Domains with Conical Points and Their Finite Element Discretisation
- Implicit Runge–Kutta Schemes for Optimal Control Problems with Evolution Equations
- A Multilevel Extension of the GDSW Overlapping Schwarz Preconditioner in Two Dimensions
- A Numerical Assessment of Finite Element Discretizations for Convection-Diffusion-Reaction Equations Satisfying Discrete Maximum Principles
- Robust Finite Element Discretization and Solvers for Distributed Elliptic Optimal Control Problems
- Finite Element Approximations for PDEs with Irregular Dirichlet Boundary Data on Boundary Concentrated Meshes