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Math."],"published-print":{"date-parts":[[2024,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside\/\u201con top\u201d of it. Here the overlapping mesh is prescribed by a simple continuous motion, meaning that its location as a function of time is <jats:italic>continuous<\/jats:italic> and <jats:italic>piecewise linear<\/jats:italic>. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche\u2019s method and also includes an integral term over the space-time boundary between the two meshes that mimics the standard discontinuous Galerkin time-jump term. The simple continuous mesh motion results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore employ what seems to be a relatively uncommon energy analysis framework for finite element methods for parabolic problems that is general and robust enough to be applicable to the current setting. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.\n<\/jats:p>","DOI":"10.1007\/s00211-024-01417-8","type":"journal-article","created":{"date-parts":[[2024,5,25]],"date-time":"2024-05-25T13:01:57Z","timestamp":1716642117000},"page":"1015-1054","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Space-time CutFEM on overlapping meshes I: simple continuous mesh motion"],"prefix":"10.1007","volume":"156","author":[{"given":"Mats G.","family":"Larson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anders","family":"Logg","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carl","family":"Lundholm","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,5,25]]},"reference":[{"key":"1417_CR1","doi-asserted-by":"crossref","unstructured":"Nitsche, J.: \u00dcber ein Variationsprinzip zur L\u00f6sung von Dirichlet-Problemen bei Verwendung von Teilr\u00e4umen, die keinen Randbedingungen unterworfen sind. 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