{"id":39545,"date":"2020-12-17T10:56:22","date_gmt":"2020-12-17T10:56:22","guid":{"rendered":"http:\/\/tugraztestweb.asol.at\/gesamtverzeichnis\/unkategorisiert\/inf-sup-stable-space-time-methods-for-time-dependent-partial-differential-equations\/"},"modified":"2020-12-17T13:57:23","modified_gmt":"2020-12-17T11:57:23","slug":"inf-sup-stable-space-time-methods-for-time-dependent-partial-differential-equations","status":"publish","type":"product","link":"https:\/\/tugraztestweb.asol.at\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/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":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v16.1.1 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<link rel=\"canonical\" href=\"https:\/\/tugraztestweb.asol.at\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/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=\"In Raum-Zeit-Methoden interpretiert man die Zeitrichtung als zus\u00e4tzliche Ortskoordinate. Der Fokus dieser Arbeit liegt auf Raum-Zeit-Variationsformulierungen f\u00fcr die W\u00e4rmeleitungsgleichung und Wellengleichung, welche entsprechende inf-sup-Bedingungen erf\u00fcllen und deren konforme Raum-Zeit-Diskretisierungen zu unbedingt stabilen Verfahren f\u00fchren.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.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.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.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/tugraztestweb.asol.at\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/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=\"2020-12-17T11:57:23+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/tugraztestweb.asol.at\/wp-content\/uploads\/2020\/12\/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<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\":\"WebSite\",\"@id\":\"https:\/\/tugraztestweb.asol.at\/#website\",\"url\":\"https:\/\/tugraztestweb.asol.at\/\",\"name\":\"Verlag der TU Graz\",\"description\":\"Verlag der Technischen Universit\\u00e4t Graz\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":\"https:\/\/tugraztestweb.asol.at\/?s={search_term_string}\",\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/tugraztestweb.asol.at\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/inf-sup-stable-space-time-methods-for-time-dependent-partial-differential-equations\/#primaryimage\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/tugraztestweb.asol.at\/wp-content\/uploads\/2020\/12\/image-978-3-85125-721-2.png\",\"contentUrl\":\"https:\/\/tugraztestweb.asol.at\/wp-content\/uploads\/2020\/12\/image-978-3-85125-721-2.png\",\"width\":592,\"height\":857},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/tugraztestweb.asol.at\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/inf-sup-stable-space-time-methods-for-time-dependent-partial-differential-equations\/#webpage\",\"url\":\"https:\/\/tugraztestweb.asol.at\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/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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