{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:56:36Z","timestamp":1776840996429,"version":"3.51.2"},"reference-count":46,"publisher":"American Mathematical Society (AMS)","issue":"358","license":[{"start":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T00:00:00Z","timestamp":1771027200000},"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-2309245"],"award-info":[{"award-number":["DMS-2309245"]}],"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-1937254"],"award-info":[{"award-number":["DMS-1937254"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We present a study on asymptotically compatible Galerkin discretizations for a class of parametrized nonlinear variational problems. The abstract analytical framework is based on variational convergence, or Gamma-convergence. We demonstrate the broad applicability of the theoretical framework by developing asymptotically compatible finite element discretizations of some representative nonlinear nonlocal variational problems on a bounded domain. These include nonlocal nonlinear problems with classically-defined, local boundary constraints through heterogeneous localization at the boundary, as well as nonlocal problems posed on parameter-dependent domains.<\/p>","DOI":"10.1090\/mcom\/4064","type":"journal-article","created":{"date-parts":[[2024,12,18]],"date-time":"2024-12-18T10:19:34Z","timestamp":1734517174000},"page":"589-612","source":"Crossref","is-referenced-by-count":2,"title":["Asymptotically compatible schemes for nonlinear variational models via Gamma-convergence and applications to nonlocal problems"],"prefix":"10.1090","volume":"95","author":[{"given":"Qiang","family":"Du","sequence":"first","affiliation":[]},{"given":"James","family":"Scott","sequence":"additional","affiliation":[]},{"given":"Xiaochuan","family":"Tian","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2025,2,14]]},"reference":[{"issue":"217","key":"1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0025-5718-97-00785-0","article-title":"Locking-free finite element methods for shells","volume":"66","author":"Arnold, Douglas N.","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"2","series-title":"Advances in Applied Mathematics","isbn-type":"print","volume-title":"Handbook of peridynamic modeling","year":"2017","ISBN":"https:\/\/id.crossref.org\/isbn\/9781482230437"},{"key":"3","isbn-type":"print","volume-title":"Optimal control and partial differential equations","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/1586030965"},{"key":"4","series-title":"Oxford Lecture Series in Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198507840.001.0001","volume-title":"$\\Gamma$-convergence for beginners","volume":"22","author":"Braides, Andrea","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0198507844"},{"key":"5","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387759333","edition":"3"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01395985","article-title":"Finite-dimensional approximation of nonlinear problems. I. Branches of nonsingular solutions","volume":"36","author":"Brezzi, F.","year":"1980","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"7","series-title":"Classics in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719208","volume-title":"The finite element method for elliptic problems","volume":"40","author":"Ciarlet, Philippe G.","year":"2002","ISBN":"https:\/\/id.crossref.org\/isbn\/0898715148"},{"key":"8","isbn-type":"print","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1007\/978-3-031-04496-0_8","article-title":"On the prescription of boundary conditions for nonlocal Poisson\u2019s and peridynamics models","author":"D\u2019Elia, Marta","year":"[2022] \\copyright2022","ISBN":"https:\/\/id.crossref.org\/isbn\/9783031044953"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"14","DOI":"10.1007\/s42102-018-0004-x","article-title":"A review of benchmark experiments for the validation of peridynamics models","volume":"1","author":"Diehl, Patrick","year":"2019","journal-title":"J. Peridyn. Nonlocal Model.","ISSN":"https:\/\/id.crossref.org\/issn\/2522-896X","issn-type":"print"},{"key":"10","series-title":"CBMS-NSF Regional Conference Series in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611975628","volume-title":"Nonlocal modeling, analysis, and computation","volume":"94","author":"Du, Qiang","year":"2019","ISBN":"https:\/\/id.crossref.org\/isbn\/9781611975611"},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1142\/S0219530521500329","article-title":"Fractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localization","volume":"20","author":"Du, Qiang","year":"2022","journal-title":"Anal. Appl. (Singap.)","ISSN":"https:\/\/id.crossref.org\/issn\/0219-5305","issn-type":"print"},{"issue":"6","key":"12","doi-asserted-by":"publisher","first-page":"Paper No. 097, 31","DOI":"10.3934\/mine.2023097","article-title":"\ud835\udc3f^{\ud835\udc5d} compactness criteria with an application to variational convergence of some nonlocal energy functionals","volume":"5","author":"Du, Qiang","year":"2023","journal-title":"Math. Eng."},{"key":"13","doi-asserted-by":"publisher","first-page":"Paper No. 112209, 20","DOI":"10.1016\/j.jcp.2023.112209","article-title":"Numerical solutions for nonlocal wave equations by perfectly matched layers II: the two-dimensional case","volume":"488","author":"Du, Yu","year":"2023","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"14","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4355-5","volume-title":"Theory and practice of finite elements","volume":"159","author":"Ern, Alexandre","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0387205748"},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1007\/s42102-021-00074-x","article-title":"Convergence analysis and numerical studies for linearly elastic peridynamics with Dirichlet-type boundary conditions","volume":"5","author":"Foss, Mikil","year":"2023","journal-title":"J. Peridyn. Nonlocal Model.","ISSN":"https:\/\/id.crossref.org\/issn\/2522-896X","issn-type":"print"},{"key":"16","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1142\/9789812795557","volume-title":"Direct methods in the calculus of variations","author":"Giusti, Enrico","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/9812380434"},{"issue":"3","key":"17","doi-asserted-by":"publisher","first-page":"A1359--A1384","DOI":"10.1137\/22M1490351","article-title":"An asymptotically compatible coupling formulation for nonlocal interface problems with jumps","volume":"45","author":"Glusa, Christian","year":"2023","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"1","key":"18","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1137\/090746938","article-title":"Asymptotic analysis of upwind discontinuous Galerkin approximation of the radiative transport equation in the diffusive limit","volume":"48","author":"Guermond, Jean-Luc","year":"2010","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"19","doi-asserted-by":"publisher","first-page":"1001","DOI":"10.1137\/S0363012994262361","article-title":"Finite-dimensional approximation of a class of constrained nonlinear optimal control problems","volume":"34","author":"Gunzburger, Max D.","year":"1996","journal-title":"SIAM J. Control Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0363-0129","issn-type":"print"},{"issue":"12","key":"20","doi-asserted-by":"publisher","first-page":"2507","DOI":"10.1142\/S0218202523500549","article-title":"Nonlocal half-ball vector operators on bounded domains: Poincar\u00e9 inequality and its applications","volume":"33","author":"Han, Zhaolong","year":"2023","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"1675","DOI":"10.3934\/dcdsb.2020178","article-title":"Finite element approximation of nonlocal dynamic fracture models","volume":"26","author":"Jha, P. K.","year":"2021","journal-title":"Discrete Contin. Dyn. Syst. Ser. B","ISSN":"https:\/\/id.crossref.org\/issn\/1531-3492","issn-type":"print"},{"key":"22","doi-asserted-by":"crossref","unstructured":"P. K. Jha and R. Lipton, Finite differences and finite elements in nonlocal fracture modeling: A priori convergence rates, Handbook of Nonlocal Continuum Mechanics for Materials and Structures, Springer, 2019, 1457\u20131494.","DOI":"10.1007\/978-3-319-58729-5_44"},{"key":"23","doi-asserted-by":"publisher","first-page":"Paper No. 110919, 24","DOI":"10.1016\/j.jcp.2021.110919","article-title":"A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems","volume":"452","author":"Leng, Yu","year":"2022","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"24","doi-asserted-by":"publisher","first-page":"113264","DOI":"10.1016\/j.cma.2020.113264","article-title":"Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation","volume":"370","author":"Leng, Yu","year":"2020","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"6","key":"25","doi-asserted-by":"publisher","first-page":"Paper No. e23143, 27","DOI":"10.1002\/num.23143","article-title":"Asymptotically compatible schemes for nonlocal Ohta-Kawasaki model","volume":"40","author":"Luo, Wangbo","year":"2024","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"issue":"1","key":"26","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/s10659-013-9456-z","article-title":"Nonlocal constrained value problems for a linear peridynamic Navier equation","volume":"116","author":"Mengesha, Tadele","year":"2014","journal-title":"J. Elasticity","ISSN":"https:\/\/id.crossref.org\/issn\/0374-3535","issn-type":"print"},{"issue":"11","key":"27","doi-asserted-by":"publisher","first-page":"3999","DOI":"10.1088\/0951-7715\/28\/11\/3999","article-title":"On the variational limit of a class of nonlocal functionals related to peridynamics","volume":"28","author":"Mengesha, Tadele","year":"2015","journal-title":"Nonlinearity","ISSN":"https:\/\/id.crossref.org\/issn\/0951-7715","issn-type":"print"},{"issue":"4","key":"28","doi-asserted-by":"publisher","first-page":"795","DOI":"10.1007\/s00161-024-01299-z","article-title":"Linearization and localization of nonconvex functionals motivated by nonlinear peridynamic models","volume":"36","author":"Mengesha, Tadele","year":"2024","journal-title":"Contin. Mech. Thermodyn.","ISSN":"https:\/\/id.crossref.org\/issn\/0935-1175","issn-type":"print"},{"key":"29","doi-asserted-by":"publisher","first-page":"Paper No. 111351, 23","DOI":"10.1016\/j.jcp.2022.111351","article-title":"Accurate absorbing boundary conditions for two-dimensional peridynamics","volume":"466","author":"Pang, Gang","year":"2022","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"3","key":"30","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s00526-003-0195-z","article-title":"A new approach to Sobolev spaces and connections to \u0393-convergence","volume":"19","author":"Ponce, Augusto C.","year":"2004","journal-title":"Calc. Var. Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0944-2669","issn-type":"print"},{"issue":"4","key":"31","doi-asserted-by":"publisher","first-page":"923","DOI":"10.1002\/num.20249","article-title":"Numerical solution of parametrized Navier-Stokes equations by reduced basis methods","volume":"23","author":"Quarteroni, Alfio","year":"2007","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"issue":"3","key":"32","doi-asserted-by":"publisher","first-page":"4185","DOI":"10.1137\/23M1588111","article-title":"Nonlocal problems with local boundary conditions I: Function spaces and variational principles","volume":"56","author":"Scott, James M.","year":"2024","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"3","key":"33","doi-asserted-by":"publisher","first-page":"4185","DOI":"10.1137\/23M1588111","article-title":"Nonlocal problems with local boundary conditions I: Function spaces and variational principles","volume":"56","author":"Scott, James M.","year":"2024","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"1","key":"34","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":"2","key":"35","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"},{"key":"36","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.cma.2017.03.043","article-title":"Stability of peridynamic correspondence material models and their particle discretizations","volume":"322","author":"Silling, S. A.","year":"2017","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"3","key":"37","doi-asserted-by":"publisher","first-page":"1052","DOI":"10.1137\/18M1184576","article-title":"Nonlocal models with heterogeneous localization and their application to seamless local-nonlocal coupling","volume":"17","author":"Tao, Yunzhe","year":"2019","journal-title":"Multiscale Model. Simul.","ISSN":"https:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"issue":"6","key":"38","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":"39","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":"40","doi-asserted-by":"publisher","first-page":"1621","DOI":"10.1137\/16M1078811","article-title":"Trace theorems for some nonlocal function spaces with heterogeneous localization","volume":"49","author":"Tian, Xiaochuan","year":"2017","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"1","key":"41","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1137\/19M1296720","article-title":"Asymptotically compatible schemes for robust discretization of parametrized problems with applications to nonlocal models","volume":"62","author":"Tian, Xiaochuan","year":"2020","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"key":"42","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/j.cma.2018.08.016","article-title":"An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics","volume":"343","author":"Trask, Nathaniel","year":"2019","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"334","key":"43","doi-asserted-by":"publisher","first-page":"761","DOI":"10.1090\/mcom\/3685","article-title":"Stability and error analysis for a second-order approximation of 1D nonlocal Schr\u00f6dinger equation under DtN-type boundary conditions","volume":"91","author":"Wang, Jihong","year":"2022","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"44","doi-asserted-by":"publisher","first-page":"1373","DOI":"10.1051\/m2an\/2019089","article-title":"An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems","volume":"54","author":"You, Huaiqian","year":"2020","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/2822-7840","issn-type":"print"},{"key":"45","doi-asserted-by":"publisher","first-page":"Paper No. 113691, 36","DOI":"10.1016\/j.cma.2021.113691","article-title":"An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture","volume":"377","author":"Yu, Yue","year":"2021","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"46","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1016\/j.apnum.2017.11.007","article-title":"An accurate and asymptotically compatible collocation scheme for nonlocal diffusion problems","volume":"133","author":"Zhang, Xiaoping","year":"2018","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-358\/S0025-5718-2025-04064-X\/mcom4064_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-358\/S0025-5718-2025-04064-X\/S0025-5718-2025-04064-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:57:59Z","timestamp":1776837479000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-358\/S0025-5718-2025-04064-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,14]]},"references-count":46,"journal-issue":{"issue":"358","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0025-5718-2025-04064-X"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4064","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":[[2025,2,14]]}}}