Free Access
Volume 54, Number 1, January-February 2020
Page(s) 105 - 128
Published online 14 January 2020
  1. F. Andreu and J. Mazón, J. Rossi and J. Toledo, Nonlocal diffusion problems. , In: Vol. 165 of Mathematical Surveys and Monographs. American Mathematical Society (2010). [CrossRef] [Google Scholar]
  2. G. Aubert and P. Kornprobst, Can the nonlocal characterization of Sobolev spaces by Bourgain et al. be useful for solving variational problems?. SIAM J. Numer. Anal. 47 (2009) 844–860. [Google Scholar]
  3. G. Barles, E. Chasseigne and C. Imbert, On the Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57 (2008) 213–246. [CrossRef] [MathSciNet] [Google Scholar]
  4. J. Bourgain, H. Brezis and P. Mironescu. Another Look at Sobolev Spaces, IOS Press, Amsterdam (2001) 439–455. [Google Scholar]
  5. A. Buades, B. Coll and J. Morel, Image denoising methods. A new nonlocal principle. SIAM Rev. 52 (2010) 113–147. [CrossRef] [Google Scholar]
  6. C. Bucur and E. Valdinoci, Nonlocal diffusion and applications. In: Vol. 20 of Lecture Notes of the Unione Matematica Italiana. Springer (2016). [CrossRef] [Google Scholar]
  7. D. Burago, S. Ivanov and Y. Kurylev, A graph discretization of the Laplace-Beltrami operator. J. Spectral Theory 4 (2014) 675–714. [CrossRef] [Google Scholar]
  8. N. Burch and R.B. Lehoucq, Classical, nonlocal, and fractional diffusion equations on bounded domains. Int. J. Multiscale Comput. Eng. 9 (2011) 661. [Google Scholar]
  9. R. Coifman and S. Lafon. Diffusion maps. Appl. Comput. Harmonic Anal. 21 (2006) 5–30. [CrossRef] [Google Scholar]
  10. S. Conti, G. Dolzmann and S. Müller, The div-curl lemma for sequences whose divergence and curl are compact in W1,1. C.R. Math. 349 (2011) 175–178. [CrossRef] [Google Scholar]
  11. O. Defterli, M. D’Elia, Q. Du, M. Gunzburger, R. Lehoucq and M. Meerschaert, Fractional diffusion on bounded domains. Fract. Calc. Appl. Anal. 18 (2015) 342–360. [Google Scholar]
  12. P. Degond and S. Mas-Gallic, The weighted particle method for convection-diffusion equations. I. The case of an isotropic viscosity. Math. Comput. 53 (1989) 485–507. [Google Scholar]
  13. Q. Du, Nonlocal modeling, analysis and computation. In: Vol. 94 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM (2019). [Google Scholar]
  14. Q. Du and X. Tian, Stability of nonlocal Dirichlet integrals and implications for peridynamic correspondence material modeling. SIAM J. Appl. Math. 78 (2018) 1536–1552. [Google Scholar]
  15. Q. Du and X. Tian, Mathematics of smoothed particle hydrodynamics, a study via nonlocal Stokes equations. To appear in: Found. Comput. Math., DOI: 10.1007/s10208-019-09432-0 (2019). [Google Scholar]
  16. Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal continuum theory. ESAIM: M2AN 45 (2011) 217–234. [CrossRef] [EDP Sciences] [Google Scholar]
  17. Q. Du, M. Gunzburger, R.B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54 (2012) 667–696. [CrossRef] [MathSciNet] [Google Scholar]
  18. Q. Du, M. Gunzburger, R.B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23 (2013) 493–540. [Google Scholar]
  19. Q. Du, Y. Tao, X. Tian and J. Yang, Robust a posteriori stress analysis for quadrature collocation approximations of nonlocal models via nonlocal gradients. Comput. Methods Appl. Mech. Eng. 310 (2016) 605–627. [Google Scholar]
  20. Q. Du, J. Yang and Z. Zhou, Analysis of a nonlocal-in-time parabolic equation. Discrete Continuous Dyn. Syst. B 22 (2017) 339–368. [CrossRef] [Google Scholar]
  21. Q. Du, T. Mengesha and X. Tian, Nonlocal criteria for compactness in the space of lp vector fields, Preprint arXiv:1801.08000 (2018) . [Google Scholar]
  22. L.C. Evans, Weak convergence methods for nonlinear partial differential equations. In: Number 74 of CBMS Regional Conference Series in Mathematics, American Mathematical Soc. (1990). [Google Scholar]
  23. M. Fuentes, M. Kuperman and V. Kenkre, Nonlocal interaction effects on pattern formation in population dynamics. Phys. Rev. Lett. 91 (2003) 158104. [CrossRef] [PubMed] [Google Scholar]
  24. M. Felsinger, M. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators. Math. Z. 279 (2015) 779–809. [CrossRef] [Google Scholar]
  25. G. Gilboa and S. Osher, Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7 (2008) 1005–1028. [Google Scholar]
  26. R. Gingold and J. Monaghan, Smoothed particle hydrodynamics: theory and application to non-spherical stars. MNRAS 181 (1977) 375–389. [NASA ADS] [CrossRef] [Google Scholar]
  27. T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius. Discrete Continuous Dyn. Syst. Ser. B 7 (2007) 125. [Google Scholar]
  28. C.-Y. Kao, Y. Lou and W. Shen, Random dispersal vs. nonlocal dispersal. Discrete Continuous. Dyn. Syst. 26 (2010) 551–596. [Google Scholar]
  29. A. Korn, Ubereinige ungleichungen, welche in der theorie der elastischen und elektrischen schwingungen eine rolle spielen. Bull. Int. Cracovie Akademie Umiejet Classe Sci. Math. Nat. 3 (1909) 705–724. [Google Scholar]
  30. M. Křížek and P. Neittaanmäki, On the validity of friedrichs’inequalities. Math. Scand. 54 (1984) 17–26. [CrossRef] [Google Scholar]
  31. H. Lee and Q. Du, Asymptotically compatible sph-like particle discretizations of one dimensional linear advection models. SIAM J. Numer. Anal. 57 (2019) 127–147. [Google Scholar]
  32. R.B. Lehoucq and S.A. Silling, Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 56 (2008) 1566–1577. [Google Scholar]
  33. X.H. Li and J. Lu, Quasi-nonlocal coupling of nonlocal diffusions. SIAM J. Numer. Anal. 55 (2017) 2394–2415. [Google Scholar]
  34. L.B. Lucy, A numerical approach to the testing of the fission hypothesis. Astron. J. 82 (1977) 1013–1024. [Google Scholar]
  35. K. Mazowiecka and A. Schikorra, Fractional div-curl quantities and applications to nonlocal geometric equations. J. Funct. Anal. 275 (2018) 1–44. [Google Scholar]
  36. T. Mengesha, Nonlocal Korn-type characterization of Sobolev vector fields. Commun. Contemp. Math. 14 (2012) 1250028. [CrossRef] [Google Scholar]
  37. T. Mengesha and Q. Du, The bond-based peridynamic system with dirichlet-type volume constraint. Proc. R. Soc. Edinburgh Sect. A: Math. 144 (2014) 161–186. [CrossRef] [Google Scholar]
  38. T. Mengesha and Q. Du, Characterization of function spaces of vector fields and an application in nonlinear peridynamics. Nonlinear Anal. 140 (2016) 82–111. [CrossRef] [Google Scholar]
  39. T. Mengesha and D. Spector, Localization of nonlocal gradients in various topologies. Calculus Variations Partial Differ. Equ. 52 (2015) 253–279. [CrossRef] [Google Scholar]
  40. F. Murat, Compacité par compensation. Anal. Scuola Normale Superiore Pisa-Classe Sci. 5 (1978) 489–507. [Google Scholar]
  41. J. Necas and I. Hlavácek, Mathematical Theory of Elastic and Elasto-plastic Bodies: An Introduction. In Vol. 3. . Elsevier (2017). [Google Scholar]
  42. S. Nugent and H.A. Posch, Liquid drops and surface tension with smoothed particle applied mechanics. Phys. Rev. E 62 (2000) 4968. [Google Scholar]
  43. J. Peddieson, G.R. Buchanan and R.P. McNitt, Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41 (2003) 305–312. [Google Scholar]
  44. L. Pismen, Nonlocal diffuse interface theory of thin films and the moving contact line. Phys. Rev. E 64 (2001) 021603. [Google Scholar]
  45. W. Radu and K. Wells, A doubly nonlocal laplace operator and its connection to the classical laplacian. J. Integral Equ. Appl. 31 (2019) 379–409. [CrossRef] [Google Scholar]
  46. J. Saranen, On an inequality of friedrichs. Math. Scand. 51 (1983) 310–322. [CrossRef] [Google Scholar]
  47. S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mechan. Phys. Solids 48 (2000) 175–209. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
  48. S.A. Silling, Stability of peridynamic correspondence material models and their particle discretizations. Comput. Methods App. Mech. Eng. 322 (2017) 42–57. [CrossRef] [Google Scholar]
  49. V. Tarasov, Fractional vector calculus and fractional Maxwell’s equations. Ann. Phys. 323 (2008) 2756–2778. [Google Scholar]
  50. R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis. In Vol. 343. American Mathematical Soc. (2001). [Google Scholar]
  51. X. Tian and Q. Du, Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J. Numer. Anal. 51 (2013) 3458–3482. [Google Scholar]
  52. H. Tian, L. Ju and Q. Du, A conservative nonlocal convection-diffusion model and asymptotically compatible finite difference discretization. Comput. Methods Appl. Mech. Eng. 320 (2017) 46–67. [Google Scholar]
  53. C.M. Topaz, A.L. Bertozzi and M.A. Lewis, A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68 (2006) 1601. [Google Scholar]
  54. A.-K. Tornberg and B. Engquist, Regularization techniques for numerical approximation of pdes with singularities. J. Sci. Comput. 19 (2003) 527–552. [Google Scholar]
  55. N. Trillos and D. Slepcev, A variational approach to the consistency of spectral clustering. Appl. Comput. Harmonic Anal. (2016). [Google Scholar]
  56. Y. van Gennip and A. Bertozzi, γ-convergence of graph Ginzburg-Landau functionals. Adv. Differ. Equ. 17 (2012) 1115–1180. [Google Scholar]
  57. K. Zhou and Q. Du, Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48 (2010) 1759–1780. [Google Scholar]
  58. Y. Zhu and P.J. Fox, Smoothed particle hydrodynamics model for diffusion through porous media. Transp. Porous Media 43 (2001) 441–471. [Google Scholar]

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