2026-09-20T09:58:07Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/124245 2024-09-16 col_2142_5131 col_2142_10761 com_2142_5130 com_2142_10755 com_2142_234

Olson, Luke N Olson, Luke N Gropp, William D Fischer, Paul MacLachlan, Scott 2024-05 application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2024-09-16 without embargo terms The student, Alexey Voronin, accepted the attached license on 2024-04-09 at 13:28. The student, Alexey Voronin, submitted this Dissertation for approval on 2024-04-09 at 13:57. This Dissertation was approved for publication on 2024-04-10 at 16:19. DSpace SAF Submission Ingestion Package generated from Vireo submission #20340 on 2024-09-16 at 00:33:54 Monolithic multigrid for saddle point systems Voronin, Alexey 2024-04-10 Multigrid Algebraic Geometric Iterative Methods Preconditioning Relaxation Additive Schwarz Vanka Stokes Equation High Order Finite Elements Saddle Point Systems Parallel Mpi High Performance Computing In computational science and engineering, the discretization of coupled partial differential equations (PDEs) modeling multi-physics phenomena leads to large linear saddle-point systems. These systems encompass multiple interlinked unknowns, such as velocity, pressure, temperature, and charge, arising in simulations across domains like hydrocarbon extraction, biomedical engineering, and plasma physics. This dissertation focuses on developing a robust multigrid preconditioning framework for these systems using the Stokes equations as a representative model problem. A novel defect-correction approach is introduced for coupled systems, utilizing stable low-order re-discretizations to construct preconditioners for higher-order discretizations like Taylor-Hood and Scott-Vogelius elements. For Taylor-Hood, geometric multigrid performance is optimized through Local Fourier Analysis. Furthermore, a monolithic algebraic multigrid (AMG) method is developed, incorporating the defect-correction approach to robustly precondition these higher-order Stokes discretizations without relying on geometric information, unlike most existing approaches. To improve efficiency for high approximation orders, p-multigrid methods combining spatial and approximation order coarsening are presented. These methods outperform traditional spatial-only multigrid, especially for unstructured meshes. Different p-coarsening strategies are analyzed, and a robust approximate full-block factorization variant leveraging p-multigrid is introduced for the Scott-Vogelius discretization. Finally, patch relaxation techniques are proposed to reduce multigrid setup costs when solving sequences of related linear systems. This approach reuses patch factorizations between consecutive solves and updates only a subset of patches, minimizing overhead while maintaining fast convergence. Text eng https://hdl.handle.net/2142/124245 Copyright 2024 Alexey Voronin Ph.D. Dissertation Computer Science University of Illinois at Urbana-Champaign Computer Science