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On generalized multiscale methods for flow in complex porous media and their applications
dc.rights.license | Atribución-NoComercial 4.0 Internacional |
dc.contributor.advisor | Galvis Arrieta, Juan Carlos |
dc.contributor.author | Contreras Hernandez, Luis Fernando |
dc.contributor.author | Fernando, Luis |
dc.date.accessioned | 2023-08-04T14:36:18Z |
dc.date.available | 2023-08-04T14:36:18Z |
dc.date.issued | 2023-06-27 |
dc.identifier.uri | https://repositorio.unal.edu.co/handle/unal/84452 |
dc.description | ilustraciones, diagramas |
dc.description.abstract | In this document, the Generalized Multiscale Finite Element Method (GMsFEM) is studied, which deals with constructing multiscale spectral basis functions designed for high-contrast multiscale problems. The multiscale basis functions are built from the product of the eigenvectors, computed from a local spectral problem and a partition of unity over the study domain. The eigenvalues detect essential features of the solutions that are not captured by the initial multiscale basis functions. This document reviews the general convergence study where the error estimates are written in terms of the eigenvalues associated with the eigenvectors not used in the construction. Error analysis involves local and global norms that measure the convergence speed of the expansion of the solution in terms of local eigenvectors; this is achieved with a careful choice of the initial multiscale basis functions and the configuration of the eigenvalue problems. Two novel important numerical applications are presented: the first is the free-boundary dam problem posed on a heterogeneous high-contrast medium, where we introduce a fictitious time variable that motivates an adequate time discretization that can be understood as a fixed-point iteration. For the steady-state solution, we use the duality method to deal with the multivalued nonlinear terms involved; then, efficient approximations of pressure and saturation are calculated using the GMsFEM method. The second application is the solution of a parabolic equation. Here implementing time discretizations, such as finite differences or exponential integrators in the presence of a high contrast coefficient, it may not be practical in because each time iteration one needs the computation of matrix operators involving very large and extremely ill-conditioned sparse matrices. The GMsFEM is essential since it allows obtaining the solution of the problem more simply, allowing to combine the GMsFEM with the method of exponential integrators in time to get a good approximation of the final temporary solution. (Texto tomado de la fuente) |
dc.description.abstract | En este documento se estudia el M´etodo de Elementos Finitos Multiescala Generalizados (GMsFEM), el cual trata de la construcci´on de funciones base espectrales multiescala que est´an dise˜nadas para problemas de alto contraste. Las funciones base multiescala se construyen a partir del producto entre los vectores propios, construidos a partir de un problema espectral local y una partici´on de la unidad sobre el dominio de estudio. Los valores propios detectan caracter´ısticas importantes de las soluciones que no son capturadas por las funciones base multiescala iniciales. En este trabajo, se presenta un estudio de convergencia donde las estimaciones de error son generales, y est´an escritas en t´erminos de los valores propios asociados a los vectores propios no utilizados en la construcci´on. El an´alisis de errores implica normas locales y globales que miden la descomposici´on de la expansi´on de la soluci´on en t´erminos de vectores propios locales, esto se logra con una elecci´on cuidadosa de las funciones de base multiescala iniciales y la configuraci´on de los problemas de valores propios. Se presentan dos aplicaciones num´ericas importantes: la primera, es el problema de represa con frontera libre planteado sobre un medio heterog´eneo de alto contraste, donde introducimos una variable de tiempo ficticia que motiva una discretizaci´on de tiempo adecuada que puede entenderse como una iteraci´on de punto fijo a la soluci´on de estado estacionario, y usamos el m´etodo de dualidad para tratar con los t´erminos no lineales multivaluados involucrados; luego, se calculan aproximaciones eficientes de la presi´on y la saturaci´on usando el m´etodo GMsFEM. La segunda aplicaci´on es la soluci´on de una ecuaci´on parab´olica donde al implementar discretizaciones de tiempo como diferencias finitas o integradores exponenciales sobre un coeficiente de alto contraste, puede no ser pr´actico porque cada iteraci´on de tiempo necesita el c´alculo de operadores matriciales que involucran matrices dispersas, muy grandes y mal condicionadas; es por esto que el GMsFEM es importante ya que permite la obtenci´on de la soluci´on del problema de una forma m´as sencilla, permitiendo combinar GMsFEM con el m´etodo de integradores exponenciales en el tiempo para obtener una buena aproximaci´on de la soluci´on temporal final |
dc.format.extent | xiii, 104 páginas |
dc.format.mimetype | application/pdf |
dc.rights.uri | http://creativecommons.org/licenses/by-nc/4.0/ |
dc.subject.ddc | 510 - Matemáticas::518 - Análisis numérico |
dc.title | On generalized multiscale methods for flow in complex porous media and their applications |
dc.type | Trabajo de grado - Doctorado |
dc.type.driver | info:eu-repo/semantics/doctoralThesis |
dc.type.version | info:eu-repo/semantics/acceptedVersion |
dc.publisher.program | Bogotá - Ciencias - Doctorado en Ciencias - Matemáticas |
dc.description.degreelevel | Doctorado |
dc.description.researcharea | Numerical analysis, Partial differential equations. |
dc.identifier.instname | Universidad Nacional de Colombia |
dc.identifier.reponame | Repositorio Institucional Universidad Nacional de Colombia |
dc.identifier.repourl | https://repositorio.unal.edu.co/ |
dc.publisher.faculty | Facultad de Ciencias |
dc.publisher.place | Bogotá,Colombia |
dc.publisher.branch | Universidad Nacional de Colombia - Sede Bogotá |
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dc.rights.accessrights | info:eu-repo/semantics/openAccess |
dc.subject.lemb | Método de elementos finitos |
dc.subject.lemb | Finite element method |
dc.subject.lemb | Análisis numéricos |
dc.subject.lemb | Numerical analysis |
dc.subject.lemb | Análisis espectral |
dc.subject.lemb | Spectrum analysis |
dc.subject.proposal | Multiesca |
dc.subject.proposal | Alto contraste |
dc.subject.proposal | FEM |
dc.subject.proposal | Métodos numéricos |
dc.title.translated | Sobre métodos multiescala generalizados para flujo en medios porosos complejos y sus aplicaciones |
dc.type.coar | http://purl.org/coar/resource_type/c_db06 |
dc.type.coarversion | http://purl.org/coar/version/c_ab4af688f83e57aa |
dc.type.content | Text |
dc.type.redcol | http://purl.org/redcol/resource_type/TD |
oaire.accessrights | http://purl.org/coar/access_right/c_abf2 |
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