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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 evolves inside\/\u201con top\u201d of it. Here the overlapping mesh is prescribed by a simple discontinuous evolution, meaning that its location, size, and shape as functions of time are <jats:italic>discontinuous<\/jats:italic> and <jats:italic>piecewise constant<\/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. The simple discontinuous mesh evolution results in a space-time discretization with a slabwise product structure between space and time which allows for existing analysis methodologies to be applied with only minor modifications. We follow the analysis methodology presented by Eriksson and Johnson (SIAM J Numer Anal 28(1):43\u201377, 1991; SIAM J Numer Anal 32(3):706\u2013740, 1995). The greatest modification is the introduction of a Ritz-like \u201cshift operator\u201d that is used to obtain the discrete strong stability needed for the error analysis. The shift operator generalizes the original analysis to some methods for which the discrete subspace at one time does not lie in the space of the stiffness form at the subsequent time. The error analysis consists of 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.<\/jats:p>","DOI":"10.1007\/s00211-024-01413-y","type":"journal-article","created":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T13:01:45Z","timestamp":1716814905000},"page":"1055-1083","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Space-time CutFEM on overlapping meshes II: simple discontinuous mesh evolution"],"prefix":"10.1007","volume":"156","author":[{"given":"Mats G.","family":"Larson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carl","family":"Lundholm","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,5,27]]},"reference":[{"key":"1413_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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