{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T21:06:53Z","timestamp":1781125613306,"version":"3.54.1"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"316","license":[{"start":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T00:00:00Z","timestamp":1559088000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0014400"],"award-info":[{"award-number":["DE-SC0014400"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We develop a class of mixed finite element schemes for stationary magnetohydrodynamics (MHD) models, using the magnetic field\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold-italic upper B\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi mathvariant=\"bold-italic\">B<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbfit {B}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and the current density\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold-italic j\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi mathvariant=\"bold-italic\">j<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbfit {j}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as discretization variables. We show that Gauss\u2019s law for the magnetic field, namely\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"nabla dot bold-italic upper B equals 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"normal\">\n                              \u2207\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u22c5\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"bold-italic\">B<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\nabla \\cdot \\mathbfit {B}=0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and the energy law for the entire system are exactly preserved in the finite element schemes. Based on some new basic estimates for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H left-parenthesis normal d normal i normal v right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">d<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">i<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">v<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">H(\\mathrm {div})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    finite elements, we show that the new finite element scheme is well-posed. Furthermore, we show the existence of solutions to the nonlinear problems and the convergence of the Picard iterations and the finite element methods under some conditions.\n                  <\/p>","DOI":"10.1090\/mcom\/3341","type":"journal-article","created":{"date-parts":[[2018,2,21]],"date-time":"2018-02-21T09:48:56Z","timestamp":1519206536000},"page":"553-581","source":"Crossref","is-referenced-by-count":37,"title":["Structure-preserving finite element methods for stationary MHD models"],"prefix":"10.1090","volume":"88","author":[{"given":"Kaibo","family":"Hu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jinchao","family":"Xu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2018,5,29]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics (Amsterdam)","isbn-type":"print","volume-title":"Sobolev spaces","volume":"140","author":"Adams, Robert A.","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0120441438","edition":"2"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492906210018","article-title":"Finite element exterior calculus, homological techniques, and applications","volume":"15","author":"Arnold, Douglas N.","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/0521868157","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1090\/S0273-0979-10-01278-4","article-title":"Finite element exterior calculus: from Hodge theory to numerical stability","volume":"47","author":"Arnold, Douglas N.","year":"2010","journal-title":"Bull. 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