{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/120444"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/120444","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On the development of preconditioning the advection diffusion operators with spectral element discretization","abstract":"Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-05-01","abstract_html":"Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-05-01","abstract_has_math":false,"creators":["Lan, Yu-Hsiang"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"M.S.","degree_level":"Thesis","degree_discipline":"Computer Science","degree_department":null,"school":null,"contributors":["Fischer, Paul"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-05","date_published":"2023-05","updated_at":"2026-07-22T22:24:57Z","subjects":["Preconditioner","Advection Diffusion","Spectral Element Method","Overlapping Schwarz"],"languages":["en","eng"],"rights":["Copyright 2023 Yu-Hsiang Lan"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/120444","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Fischer, Paul"]},{"key":"dc:creator","label":"Author","values":["Lan, Yu-Hsiang"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-05","2023-05-01"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Computer Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]},{"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":["Preconditioner","Advection Diffusion","Spectral Element Method","Overlapping Schwarz"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2023 Yu-Hsiang Lan"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/120444"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-05-01","The student, Yu-Hsiang Lan, accepted the attached license on 2023-05-01 at 09:28.","The student, Yu-Hsiang Lan, submitted this Thesis for approval on 2023-05-01 at 09:39.","This Thesis was approved for publication on 2023-05-01 at 17:22.","DSpace SAF Submission Ingestion Package generated from Vireo submission #19271 on 2023-09-01 at 17:15:39","We discuss the two-level overlapping Schwarz methods as the main component to build the preconditioner of the advection diffusion operators under the spectral element discretization. During the construction of the Schwarz subdomain problems, several options are consid- ered including additive Schwarz versus multiplicative Schwarz, restricted or not, Dirichlet- Neumann (Dir-Neu) boundary conditions (BCs) and different sizes of overlap. To assemble the local subdomain matrix, we propose the mixed usage of spectral element matrix and linear finite element matrix (SF-matrix) based on the different regions of the subdomains so it can recover the exact BCs, Dir-Neu in particular. We verify our implementation with 1D examples of both Poisson problem and the advection diffusion cases. This is done by a careful derivation from theories to the algorithms and we present the numerical results in the comparison among various combinations of the options. It’s shown that Dir-Neu can indeed reduce the oscillations and the SF-matrix is the key to sustain the approximated BCs. Subdomains for 2D and 3D cases, however, are not well-defined due to the imperfect shape of subdomains with undefined corner regions. This will require extra treatments to recover the exact BCs in the future development. The second component experiments the usage of low-order finite element method (FEM) to build the global operator as the preliminary study of the coarse grid operator using 2D test cases. Among many variants of incomplete LU factorization (ILU), the Crout version seems to be able to produce stable and sparse factorization. We also find the dealiased coarse matrix improve the stability for the deformed elements and it’s needed to have good iteration number at convection dominant scenario. As there is no spectral equivalence of SEM-FEM for advection operator, the future development should focus on a better representation of the convection term especially at a coarse grid. Few directions are discussed in the final conclusion as our future works."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["On the development of preconditioning the advection diffusion operators with spectral element discretization"]}]}],"canonical_facts":{"dc:contributor":["Fischer, Paul"],"dc:creator":["Lan, Yu-Hsiang"],"dc:date":["2023-05","2023-05-01"],"dc:description":["Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2025-05-01","The student, Yu-Hsiang Lan, accepted the attached license on 2023-05-01 at 09:28.","The student, Yu-Hsiang Lan, submitted this Thesis for approval on 2023-05-01 at 09:39.","This Thesis was approved for publication on 2023-05-01 at 17:22.","DSpace SAF Submission Ingestion Package generated from Vireo submission #19271 on 2023-09-01 at 17:15:39","We discuss the two-level overlapping Schwarz methods as the main component to build the preconditioner of the advection diffusion operators under the spectral element discretization. During the construction of the Schwarz subdomain problems, several options are consid- ered including additive Schwarz versus multiplicative Schwarz, restricted or not, Dirichlet- Neumann (Dir-Neu) boundary conditions (BCs) and different sizes of overlap. To assemble the local subdomain matrix, we propose the mixed usage of spectral element matrix and linear finite element matrix (SF-matrix) based on the different regions of the subdomains so it can recover the exact BCs, Dir-Neu in particular. We verify our implementation with 1D examples of both Poisson problem and the advection diffusion cases. This is done by a careful derivation from theories to the algorithms and we present the numerical results in the comparison among various combinations of the options. It’s shown that Dir-Neu can indeed reduce the oscillations and the SF-matrix is the key to sustain the approximated BCs. Subdomains for 2D and 3D cases, however, are not well-defined due to the imperfect shape of subdomains with undefined corner regions. This will require extra treatments to recover the exact BCs in the future development. The second component experiments the usage of low-order finite element method (FEM) to build the global operator as the preliminary study of the coarse grid operator using 2D test cases. Among many variants of incomplete LU factorization (ILU), the Crout version seems to be able to produce stable and sparse factorization. We also find the dealiased coarse matrix improve the stability for the deformed elements and it’s needed to have good iteration number at convection dominant scenario. As there is no spectral equivalence of SEM-FEM for advection operator, the future development should focus on a better representation of the convection term especially at a coarse grid. Few directions are discussed in the final conclusion as our future works."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/120444"],"dc:language":["en","eng"],"dc:rights":["Copyright 2023 Yu-Hsiang Lan"],"dc:subject":["Preconditioner","Advection Diffusion","Spectral Element Method","Overlapping Schwarz"],"dc:title":["On the development of preconditioning the advection diffusion operators with spectral element discretization"],"dc:type":["text"],"thesis:degree_discipline":["Computer Science"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["M.S."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:57Z"}