{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,8,21]],"date-time":"2023-08-21T13:12:28Z","timestamp":1692623548921},"reference-count":26,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We perform the error analysis of a stabilized discontinuous Galerkin scheme for the initial boundary value problem associated with the magnetic induction equations using standard discontinuous Lagrange basis functions.\nIn order to obtain the quasi-optimal convergence incorporating second-order Runge\u2013Kutta schemes for time discretization, we need a strengthened <jats:inline-formula id=\"j_cmam-2018-0032_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mn>4<\/m:mn>\n                              <m:mo>\/<\/m:mo>\n                              <m:mn>3<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0032_eq_0313.png\" \/>\n                        <jats:tex-math>{4\/3}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-CFL condition (<jats:inline-formula id=\"j_cmam-2018-0032_ineq_9998\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mrow>\n                                 <m:mi mathvariant=\"normal\">\u0394<\/m:mi>\n                                 <m:mo>\u2062<\/m:mo>\n                                 <m:mi>t<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>\u223c<\/m:mo>\n                              <m:msup>\n                                 <m:mi>h<\/m:mi>\n                                 <m:mrow>\n                                    <m:mn>4<\/m:mn>\n                                    <m:mo>\/<\/m:mo>\n                                    <m:mn>3<\/m:mn>\n                                 <\/m:mrow>\n                              <\/m:msup>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0032_eq_0365.png\" \/>\n                        <jats:tex-math>{\\Delta t\\sim h^{4\/3}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>).\nTo overcome this unusual restriction on the CFL condition, we consider the explicit third-order Runge\u2013Kutta scheme for time discretization.\nWe demonstrate the error estimates in <jats:inline-formula id=\"j_cmam-2018-0032_ineq_9997\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>L<\/m:mi>\n                              <m:mn>2<\/m:mn>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0032_eq_0346.png\" \/>\n                        <jats:tex-math>{L^{2}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-sense and obtain quasi-optimal convergence for smooth solution in space and time for piecewise polynomials with any degree <jats:inline-formula id=\"j_cmam-2018-0032_ineq_9996\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>l<\/m:mi>\n                              <m:mo>\u2265<\/m:mo>\n                              <m:mn>1<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0032_eq_0474.png\" \/>\n                        <jats:tex-math>{l\\geq 1}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> under the standard CFL condition.<\/jats:p>","DOI":"10.1515\/cmam-2018-0032","type":"journal-article","created":{"date-parts":[[2018,9,11]],"date-time":"2018-09-11T09:01:39Z","timestamp":1536656499000},"page":"121-140","source":"Crossref","is-referenced-by-count":2,"title":["A Priori Error Analysis of a Discontinuous Galerkin Scheme for the Magnetic Induction Equation"],"prefix":"10.1515","volume":"20","author":[{"given":"Tanmay","family":"Sarkar","sequence":"first","affiliation":[{"name":"Tata Instiute of Fundamental Research , Centre for Applicable Mathematics , Post Bag No. 6503, GKVK Post Office, Sharada Nagar, Chikkabommasandra , Bangalore 560065 ; and Department of Mathematics, Indian Institute of Technology Jammu, NH-44 Bypass Road, Jagti, Jammu 181 221 , India"}]}],"member":"374","published-online":{"date-parts":[[2018,9,11]]},"reference":[{"key":"2023033110163449075_j_cmam-2018-0032_ref_001","doi-asserted-by":"crossref","unstructured":"N.  Besse and D.  Kr\u00f6ner,\nConvergence of locally divergence-free discontinuous-Galerkin methods for the induction equations of the 2D-MHD system,\nESAIM Math. Model. Numer. Anal. 39 (2005), no. 6, 1177\u20131202.","DOI":"10.1051\/m2an:2005051"},{"key":"2023033110163449075_j_cmam-2018-0032_ref_002","doi-asserted-by":"crossref","unstructured":"J. U.  Brackbill and D. C.  Barnes,\nThe effect of nonzero \n                  \n                     \n                        \n                           \u2207\n                           \u22c5\n                           B\n                        \n                     \n                     \n                     {\\nabla\\cdot B}\n                  \n                on the numerical solution of the magnetohydrodynamic equations,\nJ. Comput. Phys. 35 (1980), no. 3, 426\u2013430.","DOI":"10.1016\/0021-9991(80)90079-0"},{"key":"2023033110163449075_j_cmam-2018-0032_ref_003","doi-asserted-by":"crossref","unstructured":"F.  Brezzi, B.  Cockburn, L. D.  Marini and E.  S\u00fcli,\nStabilization mechanisms in discontinuous Galerkin finite element methods,\nComput. Methods Appl. Mech. Engrg. 195 (2006), no. 25\u201328, 3293\u20133310.","DOI":"10.1016\/j.cma.2005.06.015"},{"key":"2023033110163449075_j_cmam-2018-0032_ref_004","doi-asserted-by":"crossref","unstructured":"E.  Burman, A.  Ern and M. A.  Fern\u00e1ndez,\nExplicit Runge\u2013Kutta schemes and finite elements with symmetric stabilization for first-order linear PDE systems,\nSIAM J. Numer. Anal. 48 (2010), no. 6, 2019\u20132042.","DOI":"10.1137\/090757940"},{"key":"2023033110163449075_j_cmam-2018-0032_ref_005","doi-asserted-by":"crossref","unstructured":"P. G.  Ciarlet,\nThe Finite Element Method for Elliptic Problems,\nStud. Math. Appl. 4,\nNorth-Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"2023033110163449075_j_cmam-2018-0032_ref_006","doi-asserted-by":"crossref","unstructured":"B.  Cockburn and C. W.  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