{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:34:17Z","timestamp":1787333657054,"version":"build-2736575974"},"reference-count":51,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2020,1]]},"abstract":"<jats:p>In this work, we propose and develop an arbitrary-order adaptive discontinuous Petrov--Galerkin (DPG) method for the nonlinear Grad--Shafranov equation. An ultraweak formulation of the DPG scheme for the equation is given based on a minimal residual method. The DPG scheme has the advantage of providing more accurate gradients compared to conventional finite element methods, which is desired for numerical solutions to the Grad--Shafranov equation. The numerical scheme is augmented with an adaptive mesh refinement approach, and a criterion based on the residual norm in the minimal residual method is developed to achieve dynamic refinement. Nonlinear solvers for the resulting system are explored and a Picard iteration with Anderson acceleration is found to be efficient to solve the system. Finally, the proposed algorithm is implemented in parallel on MFEM using a domain-decomposition approach, and our implementation is general, supporting arbitrary order of accuracy and general meshes. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithm.<\/jats:p>","DOI":"10.1137\/19m1309894","type":"journal-article","created":{"date-parts":[[2020,10,8]],"date-time":"2020-10-08T14:12:06Z","timestamp":1602166326000},"page":"B1227-B1249","source":"Crossref","is-referenced-by-count":9,"title":["An Adaptive Discontinuous Petrov--Galerkin Method for the Grad--Shafranov Equation"],"prefix":"10.1137","volume":"42","author":[{"given":"Zhichao","family":"Peng","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Qi","family":"Tang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xian-Zhu","family":"Tang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,10,8]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1145\/321296.321305"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1088\/0741-3335\/44\/5\/304"},{"key":"atypb4","unstructured":"S. Balay et al.\n                      PETSc Users Manual\n                      , Technical Report, ANL-95\/11 Rev. 313, Argonne National Laboratory, Argonne, IL, 2019."},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/130936725"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2014.04.018"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-018-0947-5"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/130924913"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1063\/1.3328818"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/18M1193992"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.compfluid.2014.02.024"},{"key":"atypb12","unstructured":"J. Chan, L. Demkowicz, R. Moser, and N. Roberts,\n                      A Class of Discontinuous Petrov-Galerkin Methods. Part V: Solution of $1$D Burgers and Navier-Stokes Equations\n                      , ICES Report 10-25, University of Texas at Austin, Austin, TX, 2010."},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2010.01.003"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1002\/num.20640"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/100809799"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1002\/9781119176817.ecm2105"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2011.09.002"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-01818-8_6"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971200"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-47789-6_66"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2013-02721-4"},{"key":"atypb22","first-page":"284","volume":"7","author":"Grad H.","year":"1958","journal-title":"J. Nuclear Energy"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1063\/1.5006917"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2017.01.006"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/060660588"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2014.02.008"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1088\/0029-5515\/47\/7\/016"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1142\/S0129183191000512"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1201\/EBK1439810958"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2017.08.043"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.4208\/jcm.2009.27.5.013"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2015.01.015"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1016\/0010-4655(96)00046-X"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1002\/nme.4300"},{"key":"atypb35","first-page":"100002","volume":"2","author":"Nagaraj S.","year":"2019","journal-title":"J. Comput. Phys. X"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2017.05.030"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396415"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.04.002"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2013.02.045"},{"key":"atypb40","unstructured":"S. Petrides,\n                      Adaptive Multilevel Solvers for the Discontinuous Petrov-Galerkin Method with an Emphasis on High-frequency Wave Propagation Problems\n                      , Ph.D. thesis, University of Texas at Austin, Austin, TX, 2019."},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2017.06.044"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0064470"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2013.12.015"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2015.07.014"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971057.ch4"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1137\/0914028"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1088\/0029-5515\/41\/11\/309"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2018.09.013"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2020.107239"},{"key":"atypb50","first-page":"545","volume":"6","author":"Shafranov V.","year":"1958","journal-title":"Sov. Phys. JETP"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1137\/10078356X"},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2010.12.001"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1309894","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:15:31Z","timestamp":1787332531000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1309894"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":51,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["10.1137\/19M1309894"],"URL":"https:\/\/doi.org\/10.1137\/19m1309894","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1]]}}}