{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,8]],"date-time":"2026-07-08T19:08:14Z","timestamp":1783537694289,"version":"3.55.0"},"reference-count":39,"publisher":"American Mathematical Society (AMS)","issue":"315","license":[{"start":{"date-parts":[[2019,4,12]],"date-time":"2019-04-12T00:00:00Z","timestamp":1555027200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1719699"],"award-info":[{"award-number":["DMS-1719699"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["MURI Center for Material Failure Prediction Through Peridynamics"],"award-info":[{"award-number":["MURI Center for Material Failure Prediction Through Peridynamics"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["W911NF-15-1-0562"],"award-info":[{"award-number":["W911NF-15-1-0562"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1521965"],"award-info":[{"award-number":["DMS-1521965"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DE-SC0016540"],"award-info":[{"award-number":["DE-SC0016540"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2017M610749"],"award-info":[{"award-number":["2017M610749"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["U1530401"],"award-info":[{"award-number":["U1530401"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DMS-1719699"],"award-info":[{"award-number":["DMS-1719699"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["MURI Center for Material Failure Prediction Through Peridynamics"],"award-info":[{"award-number":["MURI Center for Material Failure Prediction Through Peridynamics"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["W911NF-15-1-0562"],"award-info":[{"award-number":["W911NF-15-1-0562"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DMS-1521965"],"award-info":[{"award-number":["DMS-1521965"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["DE-SC0016540"],"award-info":[{"award-number":["DE-SC0016540"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["2017M610749"],"award-info":[{"award-number":["2017M610749"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["U1530401"],"award-info":[{"award-number":["U1530401"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["DMS-1719699"],"award-info":[{"award-number":["DMS-1719699"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["MURI Center for Material Failure Prediction Through Peridynamics"],"award-info":[{"award-number":["MURI Center for Material Failure Prediction Through Peridynamics"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["W911NF-15-1-0562"],"award-info":[{"award-number":["W911NF-15-1-0562"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["DMS-1521965"],"award-info":[{"award-number":["DMS-1521965"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["DE-SC0016540"],"award-info":[{"award-number":["DE-SC0016540"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["2017M610749"],"award-info":[{"award-number":["2017M610749"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000183","name":"Army Research Office","doi-asserted-by":"publisher","award":["U1530401"],"award-info":[{"award-number":["U1530401"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1719699"],"award-info":[{"award-number":["DMS-1719699"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["MURI Center for Material Failure Prediction Through Peridynamics"],"award-info":[{"award-number":["MURI Center for Material Failure Prediction Through Peridynamics"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["W911NF-15-1-0562"],"award-info":[{"award-number":["W911NF-15-1-0562"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1521965"],"award-info":[{"award-number":["DMS-1521965"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DE-SC0016540"],"award-info":[{"award-number":["DE-SC0016540"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2017M610749"],"award-info":[{"award-number":["2017M610749"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["U1530401"],"award-info":[{"award-number":["U1530401"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DMS-1719699"],"award-info":[{"award-number":["DMS-1719699"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["MURI Center for Material Failure Prediction Through Peridynamics"],"award-info":[{"award-number":["MURI Center for Material Failure Prediction Through Peridynamics"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["W911NF-15-1-0562"],"award-info":[{"award-number":["W911NF-15-1-0562"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DMS-1521965"],"award-info":[{"award-number":["DMS-1521965"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0016540"],"award-info":[{"award-number":["DE-SC0016540"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["2017M610749"],"award-info":[{"award-number":["2017M610749"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["U1530401"],"award-info":[{"award-number":["U1530401"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["DMS-1719699"],"award-info":[{"award-number":["DMS-1719699"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["MURI Center for Material Failure Prediction Through Peridynamics"],"award-info":[{"award-number":["MURI Center for Material Failure Prediction Through Peridynamics"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["W911NF-15-1-0562"],"award-info":[{"award-number":["W911NF-15-1-0562"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["DMS-1521965"],"award-info":[{"award-number":["DMS-1521965"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["DE-SC0016540"],"award-info":[{"award-number":["DE-SC0016540"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2017M610749"],"award-info":[{"award-number":["2017M610749"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U1530401"],"award-info":[{"award-number":["U1530401"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    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\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>L<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -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.\n                  <\/p>","DOI":"10.1090\/mcom\/3333","type":"journal-article","created":{"date-parts":[[2017,11,8]],"date-time":"2017-11-08T09:39:42Z","timestamp":1510133982000},"page":"123-147","source":"Crossref","is-referenced-by-count":20,"title":["A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems"],"prefix":"10.1090","volume":"88","author":[{"given":"Qiang","family":"Du","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lili","family":"Ju","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jianfang","family":"Lu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2018,4,12]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"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.","DOI":"10.1088\/1742-6596\/125\/1\/012078"},{"key":"2","series-title":"Mathematical Surveys and Monographs","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/165","volume-title":"Nonlocal diffusion problems","volume":"165","author":"Andreu-Vaillo, Fuensanta","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821852309"},{"issue":"5-6","key":"3","doi-asserted-by":"publisher","first-page":"1119","DOI":"10.1023\/A:1004514803625","article-title":"An integrodifferential model for phase transitions: stationary solutions in higher space dimensions","volume":"95","author":"Bates, Peter W.","year":"1999","journal-title":"J. Statist. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-4715","issn-type":"print"},{"issue":"9-12","key":"4","doi-asserted-by":"publisher","first-page":"1237","DOI":"10.1016\/j.cma.2010.10.014","article-title":"Continuous and discontinuous finite element methods for a peridynamics model of mechanics","volume":"200","author":"Chen, X.","year":"2011","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"190","key":"5","doi-asserted-by":"publisher","first-page":"545","DOI":"10.2307\/2008501","article-title":"The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case","volume":"54","author":"Cockburn, Bernardo","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"6","series-title":"Studies in Mathematics and its Applications, Vol. 4","isbn-type":"print","volume-title":"The finite element method for elliptic problems","author":"Ciarlet, Philippe G.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0444850287"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1016\/0021-9991(89)90183-6","article-title":"TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems","volume":"84","author":"Cockburn, Bernardo","year":"1989","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1051\/m2an\/1991250303371","article-title":"The Runge-Kutta local projection \ud835\udc43\u00b9-discontinuous-Galerkin finite element method for scalar conservation laws","volume":"25","author":"Cockburn, Bernardo","year":"1991","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"186","key":"9","doi-asserted-by":"publisher","first-page":"411","DOI":"10.2307\/2008474","article-title":"TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework","volume":"52","author":"Cockburn, Bernardo","year":"1989","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1006\/jcph.1998.5892","article-title":"The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems","volume":"141","author":"Cockburn, Bernardo","year":"1998","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"6","key":"11","doi-asserted-by":"publisher","first-page":"2440","DOI":"10.1137\/S0036142997316712","article-title":"The local discontinuous Galerkin method for time-dependent convection-diffusion systems","volume":"35","author":"Cockburn, Bernardo","year":"1998","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"12","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1137\/110833294","article-title":"Analysis and approximation of nonlocal diffusion problems with volume constraints","volume":"54","author":"Du, Qiang","year":"2012","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1142\/S0218202512500546","article-title":"A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws","volume":"23","author":"Du, Qiang","year":"2013","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"issue":"4","key":"14","doi-asserted-by":"publisher","first-page":"961","DOI":"10.3934\/dcdsb.2014.19.961","article-title":"Nonlocal convection-diffusion volume-constrained problems and jump processes","volume":"19","author":"Du, Qiang","year":"2014","journal-title":"Discrete Contin. Dyn. Syst. Ser. B","ISSN":"https:\/\/id.crossref.org\/issn\/1531-3492","issn-type":"print"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"1899","DOI":"10.1137\/15M1039857","article-title":"Asymptotically compatible Fourier spectral approximations of nonlocal Allen-Cahn equations","volume":"54","author":"Du, Qiang","year":"2016","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"339","DOI":"10.3934\/dcdsb.2017016","article-title":"Analysis of a nonlocal-in-time parabolic equation","volume":"22","author":"Du, Qiang","year":"2017","journal-title":"Discrete Contin. Dyn. Syst. Ser. B","ISSN":"https:\/\/id.crossref.org\/issn\/1531-3492","issn-type":"print"},{"issue":"2","key":"17","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1051\/m2an\/2010040","article-title":"Mathematical analysis for the peridynamic nonlocal continuum theory","volume":"45","author":"Du, Qiang","year":"2011","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/2822-7840","issn-type":"print"},{"issue":"4","key":"18","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1177\/1081286505059748","article-title":"Analysis and numerical approximation of an integro-differential equation modeling non-local effects in linear elasticity","volume":"12","author":"Emmrich, Etienne","year":"2007","journal-title":"Math. Mech. Solids","ISSN":"https:\/\/id.crossref.org\/issn\/1081-2865","issn-type":"print"},{"key":"19","isbn-type":"print","first-page":"153","article-title":"Some nonclassical trends in parabolic and parabolic-like evolutions","author":"Fife, Paul","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/3540441980"},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1137\/060669358","article-title":"Nonlocal linear image regularization and supervised segmentation","volume":"6","author":"Gilboa, Guy","year":"2007","journal-title":"Multiscale Model. Simul.","ISSN":"https:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"1005","DOI":"10.1137\/070698592","article-title":"Nonlocal operators with applications to image processing","volume":"7","author":"Gilboa, Guy","year":"2008","journal-title":"Multiscale Model. Simul.","ISSN":"https:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"issue":"1","key":"22","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1017\/S0308210512001436","article-title":"The bond-based peridynamic system with Dirichlet-type volume constraint","volume":"144","author":"Mengesha, Tadele","year":"2014","journal-title":"Proc. Roy. Soc. Edinburgh Sect. A","ISSN":"https:\/\/id.crossref.org\/issn\/0308-2105","issn-type":"print"},{"key":"23","unstructured":"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."},{"key":"24","doi-asserted-by":"crossref","unstructured":"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\u201325.","DOI":"10.1016\/j.ijimpeng.2016.09.003"},{"key":"25","first-page":"905","article-title":"On learning with integral operators","volume":"11","author":"Rosasco, Lorenzo","year":"2010","journal-title":"J. Mach. Learn. Res.","ISSN":"https:\/\/id.crossref.org\/issn\/1532-4435","issn-type":"print"},{"issue":"2","key":"26","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/s10659-007-9125-1","article-title":"Peridynamic states and constitutive modeling","volume":"88","author":"Silling, S. A.","year":"2007","journal-title":"J. Elasticity","ISSN":"https:\/\/id.crossref.org\/issn\/0374-3535","issn-type":"print"},{"issue":"1","key":"27","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/S0022-5096(99)00029-0","article-title":"Reformulation of elasticity theory for discontinuities and long-range forces","volume":"48","author":"Silling, S. A.","year":"2000","journal-title":"J. Mech. Phys. Solids","ISSN":"https:\/\/id.crossref.org\/issn\/0022-5096","issn-type":"print"},{"issue":"1","key":"28","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1007\/s10659-009-9234-0","article-title":"Linearized theory of peridynamic states","volume":"99","author":"Silling, S. A.","year":"2010","journal-title":"J. Elasticity","ISSN":"https:\/\/id.crossref.org\/issn\/0374-3535","issn-type":"print"},{"key":"29","doi-asserted-by":"crossref","unstructured":"S. A. Silling and R. B. Lehoucq, Peridynamic theory of solid mechanics, Advances in Applied Mechanics 44 (2010), 73-168.","DOI":"10.1016\/S0065-2156(10)44002-8"},{"key":"30","doi-asserted-by":"crossref","unstructured":"S. A. Silling, O. Weckner, E. Askari, and F. Bobaru, Crack nucleation in a peridynamic solid, International Journal of Fracture 162 (2010), 219\u2013227.","DOI":"10.1007\/978-90-481-9760-6_17"},{"issue":"1-3","key":"31","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1023\/B:ELAS.0000029931.03844.4f","article-title":"Deformation of a peridynamic bar","volume":"73","author":"Silling, S. A.","year":"2003","journal-title":"J. Elasticity","ISSN":"https:\/\/id.crossref.org\/issn\/0374-3535","issn-type":"print"},{"issue":"6","key":"32","doi-asserted-by":"publisher","first-page":"3458","DOI":"10.1137\/13091631X","article-title":"Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations","volume":"51","author":"Tian, Xiaochuan","year":"2013","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"33","doi-asserted-by":"publisher","first-page":"1641","DOI":"10.1137\/130942644","article-title":"Asymptotically compatible schemes and applications to robust discretization of nonlocal models","volume":"52","author":"Tian, Xiaochuan","year":"2014","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"34","doi-asserted-by":"publisher","first-page":"762","DOI":"10.1137\/140978831","article-title":"Nonconforming discontinuous Galerkin methods for nonlocal variational problems","volume":"53","author":"Tian, Xiaochuan","year":"2015","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"35","doi-asserted-by":"publisher","first-page":"1363","DOI":"10.1007\/s10444-016-9466-z","article-title":"Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains","volume":"42","author":"Tian, Xiaochuan","year":"2016","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"36","doi-asserted-by":"publisher","first-page":"60","DOI":"10.1016\/j.cma.2015.02.008","article-title":"Nonlocal convection-diffusion problems and finite element approximations","volume":"289","author":"Tian, Hao","year":"2015","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"37","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1016\/j.cma.2017.03.020","article-title":"A conservative nonlocal convection-diffusion model and asymptotically compatible finite difference discretization","volume":"320","author":"Tian, Hao","year":"2017","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"38","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1016\/j.amc.2017.01.061","article-title":"Nonlocal diffusion and peridynamic models with Neumann type constraints and their numerical approximations","volume":"305","author":"Tao, Yunzhe","year":"2017","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"issue":"11","key":"39","doi-asserted-by":"publisher","first-page":"2497","DOI":"10.1016\/j.camwa.2015.09.030","article-title":"Generalized local and nonlocal master equations for some stochastic processes","volume":"71","author":"Zhao, Yanxiang","year":"2016","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-315\/S0025-5718-2018-03333-6\/mcom3333_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/www.ams.org\/mcom\/2019-88-315\/S0025-5718-2018-03333-6\/S0025-5718-2018-03333-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-315\/S0025-5718-2018-03333-6\/S0025-5718-2018-03333-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:54:07Z","timestamp":1776801247000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2019-88-315\/S0025-5718-2018-03333-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,12]]},"references-count":39,"journal-issue":{"issue":"315","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["S0025-5718-2018-03333-6"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3333","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2018,4,12]]}}}