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Weak Galerkin finite element methods for elasticity and coupled flow problems

dc.contributor.authorHarper, Graham Bennett, author
dc.contributor.authorLiu, Jiangguo, advisor
dc.contributor.authorBangerth, Wolfgang, committee member
dc.contributor.authorGuzik, Stephen, committee member
dc.contributor.authorTavener, Simon, committee member
dc.contributor.authorZhou, Yongcheng, committee member
dc.date.accessioned2020-08-31T10:12:11Z
dc.date.available2020-08-31T10:12:11Z
dc.date.issued2020
dc.descriptionIncludes bibliographical references.
dc.description2020 Summer.
dc.description.abstractWe present novel stabilizer-free weak Galerkin finite element methods for linear elasticity and coupled Stokes-Darcy flow with a comprehensive treatment of theoretical results and the numerical methods for each. Weak Galerkin finite element methods take a discontinuous approximation space and bind degrees of freedom together through the discrete weak gradient, which involves solving a small symmetric positive-definite linear system on every element of the mesh. We introduce notation and analysis using a general framework that highlights properties that unify many existing weak Galerkin methods. This framework makes analysis for the methods much more straightforward. The method for linear elasticity on quadrilateral and hexahedral meshes uses piecewise constant vectors to approximate the displacement on each cell, and it uses the Raviart-Thomas space for the discrete weak gradient. We use the Schur complement to simplify the solution of the global linear system and increase computational efficiency further. We prove first-order convergence in the L2 norm, verify our analysis with numerical experiments, and compare to another weak Galerkin approach for this problem. The method for coupled Stokes-Darcy flow uses an extensible multinumerics approach on quadrilateral meshes. The Darcy flow discretization uses a weak Galerkin finite element method with piecewise constants approximating pressure and the Arbogast-Correa space for the weak gradient. The Stokes domain discretization uses the classical Bernardi-Raugel pair. We prove first-order convergence in the energy norm and verify our analysis with numerical experiments. All algorithms implemented in this dissertation are publicly available as part of James Liu's DarcyLite and Darcy+ packages and as part of the deal.II library.
dc.format.mediumborn digital
dc.format.mediumdoctoral dissertations
dc.identifierHarper_colostate_0053A_16247.pdf
dc.identifier.urihttps://hdl.handle.net/10217/211829
dc.languageEnglish
dc.language.isoeng
dc.publisherColorado State University. Libraries
dc.relation.ispartof2020-
dc.rightsCopyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.
dc.subjectDarcy
dc.subjectlinear elasticity
dc.subjectweak Galerkin
dc.subjectfinite element method
dc.subjectcoupling
dc.subjectStokes
dc.titleWeak Galerkin finite element methods for elasticity and coupled flow problems
dc.typeText
dcterms.rights.dplaThis Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).
thesis.degree.disciplineMathematics
thesis.degree.grantorColorado State University
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (Ph.D.)

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