{"id":39545,"date":"2026-03-28T07:33:23","date_gmt":"2026-03-28T07:33:23","guid":{"rendered":"http:\/\/www.tugraz-verlag.at\/gesamtverzeichnis\/unkategorisiert\/inf-sup-stable-space-time-methods-for-time-dependent-partial-differential-equations\/"},"modified":"2026-03-28T11:34:23","modified_gmt":"2026-03-28T09:34:23","slug":"inf-sup-stable-space-time-methods-for-time-dependent-partial-differential-equations","status":"publish","type":"product","link":"https:\/\/www.tugraz-verlag.at\/en\/gesamtverzeichnis\/mathematik-physik-und-geodaesie\/inf-sup-stable-space-time-methods-for-time-dependent-partial-differential-equations\/","title":{"rendered":"Inf-Sup Stable Space-Time Methods for Time-Dependent Partial Differential Equations"},"content":{"rendered":"<p>For the discretisation of time-dependent partial differential equations, the classical approaches are time stepping schemes together with finite element methods in space. An alternative is to discretise the time-dependent problem without separating the temporal and spatial variables. However, space-time approximation methods depend strongly on the space-time variational formulations on the continuous level. The focus of this work is on space-time variational formulations for the heat and wave equation, which result not only in inf-sup stable formulations but fit also very well to conforming space-time discretisations.<\/p>\n<p>The first part investigates the heat equation in anisotropic Sobolev spaces, where a type of Hilbert transform is introduced such that ansatz and test spaces are equal. Unconditional stability is proven for any conforming discretisation of this space-time variational formulation.<\/p>\n<p>The second part considers space-time variational formulations for the wave equation. New existence and uniqueness results for the wave equation in a weak and in a strong sense are proven, including isomorphic solution operators and corresponding inf-sup conditions. In addition, an unconditionally stable space-time finite element method with piecewise linear, continuous functions is derived.<\/p>","protected":false},"excerpt":{"rendered":"<p>For the discretisation of time-dependent partial differential equations, the classical approaches are time stepping schemes together with finite element methods in space. An alternative is to discretise the time-dependent problem without separating the temporal and spatial variables. However, space-time approximation methods depend strongly on the space-time variational formulations on the continuous level. The focus of this work is on space-time variational formulations for the heat and wave equation, which result not only in inf-sup stable formulations but fit also very well to conforming space-time discretisations.<\/p>\n<p>The first part investigates the heat equation in anisotropic Sobolev spaces, where a type of Hilbert transform is introduced such that ansatz and test spaces are equal. Unconditional stability is proven for any conforming discretisation of this space-time variational formulation.<\/p>\n<p>The second part considers space-time variational formulations for the wave equation. New existence and uniqueness results for the wave equation in a weak and in a strong sense are proven, including isomorphic solution operators and corresponding inf-sup conditions. In addition, an unconditionally stable space-time finite element method with piecewise linear, continuous functions is derived.<\/p>\n","protected":false},"featured_media":40281,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false},"product_brand":[],"product_cat":[1630],"product_tag":[],"class_list":{"0":"post-39545","1":"product","2":"type-product","3":"status-publish","4":"has-post-thumbnail","6":"product_cat-mathematik-physik-und-geodaesie","7":"autor-marco-zank","8":"reihe-monographic-series-tu-graz","9":"reihe-monographic-series-tu-grazcomputation-in-engineering-and-science","11":"first","12":"instock","13":"taxable","14":"shipping-taxable","15":"purchasable","16":"product-type-simple"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Inf-Sup Stable Space-Time Methods for Time-Dependent Partial Differential Equations - Verlag der TU Graz<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.tugraz-verlag.at\/en\/gesamtverzeichnis\/mathematik-physik-und-geodaesie\/inf-sup-stable-space-time-methods-for-time-dependent-partial-differential-equations\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Inf-Sup Stable Space-Time Methods for Time-Dependent Partial Differential Equations - Verlag der TU Graz\" \/>\n<meta property=\"og:description\" content=\"For the discretisation of time-dependent partial differential equations, the classical approaches are time stepping schemes together with finite element methods in space. 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In addition, an unconditionally stable space-time finite element method with piecewise linear, continuous functions is derived.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.tugraz-verlag.at\/en\/gesamtverzeichnis\/mathematik-physik-und-geodaesie\/inf-sup-stable-space-time-methods-for-time-dependent-partial-differential-equations\/\" \/>\n<meta property=\"og:site_name\" content=\"Verlag der TU Graz\" \/>\n<meta property=\"article:modified_time\" content=\"2026-03-28T09:34:23+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.tugraz-verlag.at\/wp-content\/uploads\/asolmerce\/image-978-3-85125-721-2.png\" \/>\n\t<meta property=\"og:image:width\" content=\"592\" \/>\n\t<meta property=\"og:image:height\" content=\"857\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.tugraz-verlag.at\/en\/gesamtverzeichnis\/mathematik-physik-und-geodaesie\/inf-sup-stable-space-time-methods-for-time-dependent-partial-differential-equations\/\",\"url\":\"https:\/\/www.tugraz-verlag.at\/en\/gesamtverzeichnis\/mathematik-physik-und-geodaesie\/inf-sup-stable-space-time-methods-for-time-dependent-partial-differential-equations\/\",\"name\":\"Inf-Sup Stable Space-Time Methods for Time-Dependent Partial Differential Equations - 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