{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/34237"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/34237","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Conservation and efficiency in least squares finite element methods","abstract":"Two of the main aspects in the numerical solution of partial differential equations include accurate discretizations and efficient solutions of the algebraic equations. With respect to discretizations, conservation is often sought after. However, least-squares finite element methods are known to be not mass conserving when solving fluid flow problems. In this dissertation we develop mass conservative least-squares finite element methods for the Stokes and Navier-Stokes equations through the use of discontinuous finite element spaces. We formulate two divergence free formulations using both a discontinuous stream-function and a locally divergence free basis and we present a thorough numerical study of both methods. This dissertation is also concerned with the efficient solution of algebraic equations via multigrid methods. Specifically, we formulate multigrid methods for high-order H(curl) conforming finite elements. Such elements are often used in mimetic discretizations of Maxwell's equations often solved in electromagnetic applications. Efficient multigrid methods for high-order H^1 conforming finite elements and also for the lowest-order H(curl) basis have been extensively studied in recent research. We draw upon elements of both algorithms to formulate multigrid methods for high-order H(curl) finite elements for hierarchical and interpolatory type.","abstract_html":"Two of the main aspects in the numerical solution of partial differential equations include accurate discretizations and efficient solutions of the algebraic equations. With respect to discretizations, conservation is often sought after. However, least-squares finite element methods are known to be not mass conserving when solving fluid flow problems. In this dissertation we develop mass conservative least-squares finite element methods for the Stokes and Navier-Stokes equations through the use of discontinuous finite element spaces. We formulate two divergence free formulations using both a discontinuous stream-function and a locally divergence free basis and we present a thorough numerical study of both methods. This dissertation is also concerned with the efficient solution of algebraic equations via multigrid methods. Specifically, we formulate multigrid methods for high-order H(curl) conforming finite elements. Such elements are often used in mimetic discretizations of Maxwell's equations often solved in electromagnetic applications. Efficient multigrid methods for high-order H^1 conforming finite elements and also for the lowest-order H(curl) basis have been extensively studied in recent research. We draw upon elements of both algorithms to formulate multigrid methods for high-order H(curl) finite elements for hierarchical and interpolatory type.","abstract_has_math":false,"creators":["Lai, James"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Computer Science","degree_department":null,"school":null,"contributors":["Olson, Luke N.","Bochev, Pavel B.","Heath, Michael T.","Gropp, William D."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-09-18T21:07:20Z","date_published":"2012-09-18T21:07:20Z","updated_at":"2026-07-22T22:25:30Z","subjects":["finite elements","least-squares finite element methods","multigrid","high-order","discontinuous least-squares","Navier-Stokes","Stokes","edge elements","H(curl) multigrid"],"languages":["en"],"rights":["Copyright 2012 James Lai"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/34237","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Olson, Luke N.","Bochev, Pavel B.","Heath, Michael T.","Gropp, William D."]},{"key":"dc:creator","label":"Author","values":["Lai, James"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012-09-18T21:07:20Z","2012-08"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Computer Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["finite elements","least-squares finite element methods","multigrid","high-order","discontinuous least-squares","Navier-Stokes","Stokes","edge elements","H(curl) multigrid"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2012 James Lai"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/34237"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Two of the main aspects in the numerical solution of partial differential equations include accurate discretizations and efficient solutions of the algebraic equations. 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Efficient multigrid methods for high-order H^1 conforming finite elements and also for the lowest-order H(curl) basis have been extensively studied in recent research. We draw upon elements of both algorithms to formulate multigrid methods for high-order H(curl) finite elements for hierarchical and interpolatory type.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2012-07-04T16:39:56Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 2 Lai_James.pdf: 17123797 bytes, checksum: c37ec61cde3c5d82d925997d59a38e97 (MD5) Lai_James.pdf: 17123869 bytes, checksum: 7db93427d2b9a6f4669a8604a4348203 (MD5)","Made available in DSpace on 2012-09-18T21:07:20Z (GMT). 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