{"id":{"repo_id":"ghent","oai_identifier":"oai:archive.ugent.be:470416"},"canonical_url":"https://search.dev.ndltd.org/etd/ghent/oai:archive.ugent.be:470416","repository":{"repo_id":"ghent","name":"Ghent University","base_url":"https://biblio.ugent.be/oai"},"display":{"title":"On numerical methods for diffusion of electric fields in type-II superconductors","abstract":"The thesis is devoted to the study of the diffusion of the electric field in type-II superconductors in low-frequency electromagnetism. The necessity for accurate numerical methods in this research domain is increasing along with the growing number and importance of industrial applications of type-II superconductors. In Chapter 1, the basic information on superconductors is shortly summarized. The mathematical model is derived based on the power law relation between electric field E and current density J describing the nonlinear resistivity behavior of type-II superconductors. The obtained model is a nonlinear degenerate transient eddy-current problem which requires deep mathematical analysis and exact study of appropriate numerical methods. Three different versions of the problem are studied: the easiest problem involving a Lipschitz-continuous modification of the E-J relation, the non-Lipschitz but coercive model based on another modification of the power law, and the most complex problem, where the unmodified power law is considered. Different stages of the analysis are worked out for these different versions of the E-J relation. The properties of special function spaces needed for the analysis of the problem are studied in Chapter 3. Basic information on the finite element method suitable for discretization of Maxwell's equations is given in Chapter 4. In Chapter 5, the convergence of the backward Euler method is studied under the assumption that the nonlinearity is defined by the unmodified power law. We deduce the convergence of the method, carry out the error estimates and present the numerical experiments. The highlight of Chapter 5 is the generalization of the div-curl lemma. In Chapter 6 we propose a linear iteration scheme to solve a 3D stationary problem. Convergence of the method, error estimates and numerical examples are carried out. The method is stable and efficient. It is based on the fixed-point principle, which constrains its speed. The convergence of the relaxation method inspired by the articles of Jäger and Kačur is studied in Chapter 7 as preliminary to possible more extensive work in this field. In the last chapter, we propose two fully discrete methods. Their convergence is proven on the basis of the error estimates.","abstract_html":"The thesis is devoted to the study of the diffusion of the electric field in type-II superconductors in low-frequency electromagnetism. The necessity for accurate numerical methods in this research domain is increasing along with the growing number and importance of industrial applications of type-II superconductors. In Chapter 1, the basic information on superconductors is shortly summarized. The mathematical model is derived based on the power law relation between electric field E and current density J describing the nonlinear resistivity behavior of type-II superconductors. The obtained model is a nonlinear degenerate transient eddy-current problem which requires deep mathematical analysis and exact study of appropriate numerical methods. Three different versions of the problem are studied: the easiest problem involving a Lipschitz-continuous modification of the E-J relation, the non-Lipschitz but coercive model based on another modification of the power law, and the most complex problem, where the unmodified power law is considered. Different stages of the analysis are worked out for these different versions of the E-J relation. The properties of special function spaces needed for the analysis of the problem are studied in Chapter 3. Basic information on the finite element method suitable for discretization of Maxwell's equations is given in Chapter 4. In Chapter 5, the convergence of the backward Euler method is studied under the assumption that the nonlinearity is defined by the unmodified power law. We deduce the convergence of the method, carry out the error estimates and present the numerical experiments. The highlight of Chapter 5 is the generalization of the div-curl lemma. In Chapter 6 we propose a linear iteration scheme to solve a 3D stationary problem. Convergence of the method, error estimates and numerical examples are carried out. The method is stable and efficient. It is based on the fixed-point principle, which constrains its speed. The convergence of the relaxation method inspired by the articles of Jäger and Kačur is studied in Chapter 7 as preliminary to possible more extensive work in this field. In the last chapter, we propose two fully discrete methods. Their convergence is proven on the basis of the error estimates.","abstract_has_math":false,"creators":["Janiková, Edita"],"institution":"Ghent University. Faculty of Engineering","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Slodicka, Marian"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2008,"date_issued":"2008","date_published":"2008","updated_at":"2026-07-24T02:22:57Z","subjects":["Mathematics and Statistics"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://biblio.ugent.be/publication/470416","https://biblio.ugent.be/publication/470416/file/4334851"],"render_values":[{"text":"https://biblio.ugent.be/publication/470416","href":"https://biblio.ugent.be/publication/470416","code":true},{"text":"https://biblio.ugent.be/publication/470416/file/4334851","href":"https://biblio.ugent.be/publication/470416/file/4334851","code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/1854/LU-470416","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Slodicka, Marian"]},{"key":"dc:creator","label":"Author","values":["Janiková, Edita"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2008"]},{"key":"dc:publisher","label":"Institution","values":["Ghent University. 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The necessity for accurate numerical methods in this research domain is increasing along with the growing number and importance of industrial applications of type-II superconductors. In Chapter 1, the basic information on superconductors is shortly summarized. The mathematical model is derived based on the power law relation between electric field E and current density J describing the nonlinear resistivity behavior of type-II superconductors. The obtained model is a nonlinear degenerate transient eddy-current problem which requires deep mathematical analysis and exact study of appropriate numerical methods. Three different versions of the problem are studied: the easiest problem involving a Lipschitz-continuous modification of the E-J relation, the non-Lipschitz but coercive model based on another modification of the power law, and the most complex problem, where the unmodified power law is considered. Different stages of the analysis are worked out for these different versions of the E-J relation. The properties of special function spaces needed for the analysis of the problem are studied in Chapter 3. Basic information on the finite element method suitable for discretization of Maxwell's equations is given in Chapter 4. In Chapter 5, the convergence of the backward Euler method is studied under the assumption that the nonlinearity is defined by the unmodified power law. We deduce the convergence of the method, carry out the error estimates and present the numerical experiments. The highlight of Chapter 5 is the generalization of the div-curl lemma. In Chapter 6 we propose a linear iteration scheme to solve a 3D stationary problem. Convergence of the method, error estimates and numerical examples are carried out. The method is stable and efficient. It is based on the fixed-point principle, which constrains its speed. The convergence of the relaxation method inspired by the articles of Jäger and Kačur is studied in Chapter 7 as preliminary to possible more extensive work in this field. In the last chapter, we propose two fully discrete methods. Their convergence is proven on the basis of the error estimates."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["On numerical methods for diffusion of electric fields in type-II superconductors"]}]}],"canonical_facts":{"dc:contributor":["Slodicka, Marian"],"dc:creator":["Janiková, Edita"],"dc:date":["2008"],"dc:description":["The thesis is devoted to the study of the diffusion of the electric field in type-II superconductors in low-frequency electromagnetism. The necessity for accurate numerical methods in this research domain is increasing along with the growing number and importance of industrial applications of type-II superconductors. In Chapter 1, the basic information on superconductors is shortly summarized. The mathematical model is derived based on the power law relation between electric field E and current density J describing the nonlinear resistivity behavior of type-II superconductors. The obtained model is a nonlinear degenerate transient eddy-current problem which requires deep mathematical analysis and exact study of appropriate numerical methods. Three different versions of the problem are studied: the easiest problem involving a Lipschitz-continuous modification of the E-J relation, the non-Lipschitz but coercive model based on another modification of the power law, and the most complex problem, where the unmodified power law is considered. Different stages of the analysis are worked out for these different versions of the E-J relation. The properties of special function spaces needed for the analysis of the problem are studied in Chapter 3. Basic information on the finite element method suitable for discretization of Maxwell's equations is given in Chapter 4. In Chapter 5, the convergence of the backward Euler method is studied under the assumption that the nonlinearity is defined by the unmodified power law. We deduce the convergence of the method, carry out the error estimates and present the numerical experiments. The highlight of Chapter 5 is the generalization of the div-curl lemma. In Chapter 6 we propose a linear iteration scheme to solve a 3D stationary problem. Convergence of the method, error estimates and numerical examples are carried out. The method is stable and efficient. It is based on the fixed-point principle, which constrains its speed. The convergence of the relaxation method inspired by the articles of Jäger and Kačur is studied in Chapter 7 as preliminary to possible more extensive work in this field. In the last chapter, we propose two fully discrete methods. Their convergence is proven on the basis of the error estimates."],"dc:format":["application/pdf"],"dc:identifier":["https://biblio.ugent.be/publication/470416","http://hdl.handle.net/1854/LU-470416","https://biblio.ugent.be/publication/470416/file/4334851"],"dc:language":["eng"],"dc:publisher":["Ghent University. Faculty of Engineering"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:subject":["Mathematics and Statistics"],"dc:title":["On numerical methods for diffusion of electric fields in type-II superconductors"],"dc:type":["dissertation","info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]},"updated_at":"2026-07-24T02:22:57Z"}