{"id":32720,"date":"2020-08-21T12:55:17","date_gmt":"2020-08-21T12:55:17","guid":{"rendered":"http:\/\/tugraztestweb.asol.at\/gesamtverzeichnis\/unkategorisiert\/mortar-domain-decomposition-methods-for-quasilinearproblems-and-applications\/"},"modified":"2020-08-21T14:56:17","modified_gmt":"2020-08-21T12:56:17","slug":"mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications","status":"publish","type":"product","link":"https:\/\/tugraztestweb.asol.at\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications\/","title":{"rendered":"Mortar Domain Decomposition Methods for Quasilinear Problems and Applications"},"content":{"rendered":"<p> The saturated-unsaturated <br \/>flow of fl<br \/>uid (water) through a porous medium can be described by the<br \/>Richards equation which was introduced by the American physicist Lorenzo Adolph Richards in 1931.<br \/>Since the Richards equation is a highly nonlinear elliptic-parabolic partial differential equation, straight-<br \/>forward approximation methods have to be handled with care or are not applicable at all. In this work<br \/>we consider a new approach to compute the approximate solution.<br \/>In a first step, we use the primal hybrid formulation to derive a system of nonlinear equations with linear<br \/>coupling conditions. To simplify the resulting system, we apply the Kirchhoff transformation to shift<br \/>the nonlinearity of the principal part from the subdomains to the interface. After the transformation, a<br \/>coupled system with a linear principal part within the subdomains and nonlinear coupling conditions is<br \/>obtained. Solvability and uniqueness of the system are discussed.<br \/>The analogy to the discrete mortar finite element method was decisive for its application to compute<br \/>the approximate solution. We use the Newton method to solve the discrete nonlinear system. In view<br \/>efficiency, domain decomposition methods for the mortar finite element method are of special interest.<br \/>Finally we present numerical examples in two and three space dimensions.<\/p>\n<p><\/p>","protected":false},"excerpt":{"rendered":"<p>The saturated-unsaturated <br \/>flow of fl<br \/>uid (water) through a porous medium can be described by the<br \/>Richards equation which was introduced by the American physicist Lorenzo Adolph Richards in 1931.<br \/>Since the Richards equation is a highly nonlinear elliptic-parabolic partial differential equation, straight-<br \/>forward approximation methods have to be handled with care or are not applicable at all. In this work<br \/>we consider a new approach to compute the approximate solution.<br \/>In a first step, we use the primal hybrid formulation to derive a system of nonlinear equations with linear<br \/>coupling conditions. To simplify the resulting system, we apply the Kirchhoff transformation to shift<br \/>the nonlinearity of the principal part from the subdomains to the interface. After the transformation, a<br \/>coupled system with a linear principal part within the subdomains and nonlinear coupling conditions is<br \/>obtained. Solvability and uniqueness of the system are discussed.<br \/>The analogy to the discrete mortar finite element method was decisive for its application to compute<br \/>the approximate solution. We use the Newton method to solve the discrete nonlinear system. In view<br \/>efficiency, domain decomposition methods for the mortar finite element method are of special interest.<br \/>Finally we present numerical examples in two and three space dimensions.<\/p>\n","protected":false},"featured_media":40099,"comment_status":"open","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\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Mortar Domain Decomposition Methods for Quasilinear Problems and Applications - Verlag der TU Graz\" \/>\n<meta property=\"og:description\" content=\"In this work we discuss a new approach for solving the nonlinear Richards equation. This approach is based on the primal hybrid formulation and and the so called Kirchhoff transformation. To discretize the resulting nonlinear system, we use the mortar finite element method. Domain decomposition methods are used to compute the approximate solution.The saturated-unsaturated flow of fluid (water) through a porous medium can be described by theRichards equation which was introduced by the American physicist Lorenzo Adolph Richards in 1931.Since the Richards equation is a highly nonlinear elliptic-parabolic partial differential equation, straight-forward approximation methods have to be handled with care or are not applicable at all. In this workwe consider a new approach to compute the approximate solution.In a first step, we use the primal hybrid formulation to derive a system of nonlinear equations with linearcoupling conditions. To simplify the resulting system, we apply the Kirchhoff transformation to shiftthe nonlinearity of the principal part from the subdomains to the interface. After the transformation, acoupled system with a linear principal part within the subdomains and nonlinear coupling conditions isobtained. Solvability and uniqueness of the system are discussed.The analogy to the discrete mortar finite element method was decisive for its application to computethe approximate solution. We use the Newton method to solve the discrete nonlinear system. In viewefficiency, domain decomposition methods for the mortar finite element method are of special interest.Finally we present numerical examples in two and three space dimensions.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/tugraztestweb.asol.at\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications\/\" \/>\n<meta property=\"og:site_name\" content=\"Verlag der TU Graz\" \/>\n<meta property=\"article:modified_time\" content=\"2020-08-21T12:56:17+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/tugraztestweb.asol.at\/wp-content\/uploads\/2020\/08\/image-978-3-85125-522-5.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"591\" \/>\n\t<meta property=\"og:image:height\" content=\"858\" \/>\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\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications\/#primaryimage\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/tugraztestweb.asol.at\/wp-content\/uploads\/2020\/08\/image-978-3-85125-522-5.jpg\",\"contentUrl\":\"https:\/\/tugraztestweb.asol.at\/wp-content\/uploads\/2020\/08\/image-978-3-85125-522-5.jpg\",\"width\":591,\"height\":858},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/tugraztestweb.asol.at\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications\/#webpage\",\"url\":\"https:\/\/tugraztestweb.asol.at\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications\/\",\"name\":\"Mortar Domain Decomposition Methods for Quasilinear Problems and Applications - Verlag der TU Graz\",\"isPartOf\":{\"@id\":\"https:\/\/tugraztestweb.asol.at\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/tugraztestweb.asol.at\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications\/#primaryimage\"},\"datePublished\":\"2020-08-21T12:55:17+00:00\",\"dateModified\":\"2020-08-21T12:56:17+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/tugraztestweb.asol.at\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/tugraztestweb.asol.at\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/tugraztestweb.asol.at\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"item\":{\"@type\":\"WebPage\",\"@id\":\"https:\/\/tugraztestweb.asol.at\/\",\"url\":\"https:\/\/tugraztestweb.asol.at\/\",\"name\":\"Home\"}},{\"@type\":\"ListItem\",\"position\":2,\"item\":{\"@type\":\"WebPage\",\"@id\":\"https:\/\/tugraztestweb.asol.at\/gesamtverzeichnis\/\",\"url\":\"https:\/\/tugraztestweb.asol.at\/gesamtverzeichnis\/\",\"name\":\"Gesamtverzeichnis\"}},{\"@type\":\"ListItem\",\"position\":3,\"item\":{\"@type\":\"WebPage\",\"@id\":\"https:\/\/tugraztestweb.asol.at\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications\/\",\"url\":\"https:\/\/tugraztestweb.asol.at\/en\/gesamtverzeichnis\/technische-mathematik-und-technische-physik\/mortar-domain-decomposition-methods-for-quasilinear-problems-and-applications\/\",\"name\":\"Mortar Domain Decomposition Methods for Quasilinear Problems and Applications\"}}]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","_links":{"self":[{"href":"https:\/\/tugraztestweb.asol.at\/en\/wp-json\/wp\/v2\/product\/32720"}],"collection":[{"href":"https:\/\/tugraztestweb.asol.at\/en\/wp-json\/wp\/v2\/product"}],"about":[{"href":"https:\/\/tugraztestweb.asol.at\/en\/wp-json\/wp\/v2\/types\/product"}],"replies":[{"embeddable":true,"href":"https:\/\/tugraztestweb.asol.at\/en\/wp-json\/wp\/v2\/comments?post=32720"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/tugraztestweb.asol.at\/en\/wp-json\/wp\/v2\/media\/40099"}],"wp:attachment":[{"href":"https:\/\/tugraztestweb.asol.at\/en\/wp-json\/wp\/v2\/media?parent=32720"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}