dc.contributor | López Rodríguez, Bibiana | |
dc.contributor | Acevedo Martínez, Ramiro Miguel | |
dc.contributor | Universidad Nacional de Colombia - Sede Medellín | |
dc.creator | Gómez Mosquera, Christian Camilo | |
dc.date.accessioned | 2020-08-19T19:24:51Z | |
dc.date.available | 2020-08-19T19:24:51Z | |
dc.date.created | 2020-08-19T19:24:51Z | |
dc.date.issued | 2020-08-13 | |
dc.identifier | https://repositorio.unal.edu.co/handle/unal/78086 | |
dc.description.abstract | The aim of this work is to show an abstract framework to analyze the family of linear degenerate parabolic problems and family of linear degenerate parabolic mixed problems. To linear degenerate parabolic mixed equations, we deduce sufficient conditions to existence and uniqueness of solution by combining the theory for the degenerate parabolic equations and the classical Babuska-Brezzi theory.
The numerical approximation was made through the finite element method in space and a Backward-Euler scheme in time. To degenerate parabolic and degenerate parabolic mixed problems, we obtain sufficient conditions to ensure that the fully-discrete problem has a unique solution and to prove quasi-optimal error estimates for the approximation. Moreover, we present a degenerate parabolic problem which arises from electromagnetic applications and deduce its well-posedness and convergence by using the developed abstract theory, including numerical tests to illustrate the performance of the method and confirm the theoretical results.
Finally, we present the linear degenerate parabolic mixed (0 g) equations. We deduce that the fully-discrete problem has a unique solution and prove quasi-optimal error estimates for the approximation. | |
dc.description.abstract | El objetivo de este trabajo es mostrar un análisis numérico abstracto para una familia de problemas parabólicos degenerados lineales y una familia de problemas parabólicos degenerados en forma mixta lineales. En los problemas parabólicos degenerados ya se conocen resultados de existencia y unicidad, por lo cual se realizan algunos detalles de las demostraciones de los mismos por ilustración, mientras que en los problemas parabólicos degenerados en forma mixta se presenta un marco teórico, se demuestra un teorema de existencia, unicidad y dependencia continua de los datos, esto es, buen planteamiento del problema.
Para la aproximación numérica de las soluciones de ambos problemas parabólicos, se propone en la variable espacial un método de elementos finitos y en la variable temporal el método de Euler implícito. Se demuestran resultados de existencia y unicidad de los esquemas totalmente discretos propuestos y bajo ciertas suposiciones de regularidad, se obtienen estimaciones del error que sugieren obtener ordenes óptimos de convergencia de los esquemas. En ambos casos se presenta modelos de aplicación que caen dentro del marco teórico estudiado, provenientes del modelo de corrientes inducidas. Por \'ultimo, se presentan experimentos numéricos que permiten confirmar los resultados teóricos obtenidos.
Finalmente, se realiza un estudio de aproximación de problemas parabólicos degenerados mixtos en la forma (0 g) en donde se demuestra que el esquema totalmente discreto propuesto tiene única solución y bajo el supuesto que el problema continuo tiene solución, se realizan estimaciones del error. | |
dc.language | spa | |
dc.publisher | Medellín - Ciencias - Doctorado en Ciencias - Matemáticas | |
dc.publisher | Escuela de matemáticas | |
dc.publisher | Universidad Nacional de Colombia - Sede Medellín | |
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dc.rights | Atribución-NoComercial 4.0 Internacional | |
dc.rights | Acceso abierto | |
dc.rights | http://creativecommons.org/licenses/by-nc/4.0/ | |
dc.rights | info:eu-repo/semantics/openAccess | |
dc.rights | Derechos reservados - Universidad Nacional de Colombia | |
dc.title | Ecuaciones parabólicas degeneradas en forma mixta | |
dc.type | Otro | |