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The mesh movement is prescribed by a discretized dynamical system for the nodal points. This system is related to the velocity and diffusion coefficient in the convection\u2013diffusion equation such that the nodal points follow the convective flow of the model. It is shown that under a restriction of the time step size the mesh movement cannot lead to an overlap of the elements and therefore an invalid mesh. Using a piecewise linear approximation, optimal error estimates in the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell ^\\infty (L^2) \\cap \\ell ^2(H_0^1)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mi>\u2113<\/mml:mi>\n                      <mml:mi>\u221e<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msup>\n                        <mml:mi>L<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:msup>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2229<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>\u2113<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msubsup>\n                        <mml:mi>H<\/mml:mi>\n                        <mml:mn>0<\/mml:mn>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msubsup>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> norm are proved in case of both, a first-order backward Euler method and a second-order two-step method in time. These results are based on new estimates of the time dependent interpolation operator derived in this work. Preservation of the total mass is verified for both choices of the time discretization. Numerical experiments are presented that confirm the error estimates and demonstrate that the proposed moving mesh scheme can circumvent limitations that the Lagrange\u2013Galerkin method on a fixed mesh exhibits.<\/jats:p>","DOI":"10.1007\/s10915-024-02673-4","type":"journal-article","created":{"date-parts":[[2024,9,27]],"date-time":"2024-09-27T15:02:44Z","timestamp":1727449364000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Error Estimates for First- and Second-Order Lagrange\u2013Galerkin Moving Mesh Schemes for the One-Dimensional Convection\u2013Diffusion Equation"],"prefix":"10.1007","volume":"101","author":[{"given":"Kharisma Surya","family":"Putri","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tatsuki","family":"Mizuochi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0319-8748","authenticated-orcid":false,"given":"Niklas","family":"Kolbe","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hirofumi","family":"Notsu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,9,27]]},"reference":[{"issue":"1","key":"2673_CR1","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1016\/0021-9991(90)90003-J","volume":"91","author":"S Acharya","year":"1990","unstructured":"Acharya, S., Moukalled, F.H.: An adaptive grid solution procedure for convection\u2013diffusion problems. 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