Monolithic multigrid for saddle point systems

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Computer Science

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2024

Author dc:creator

• Voronin, Alexey

Contributors dc:contributor

• Olson, Luke N
• Gropp, William D
• Fischer, Paul
• MacLachlan, Scott

• Multigrid
• Algebraic
• Geometric
• Iterative Methods
• Preconditioning
• Relaxation
• Additive Schwarz
• Vanka
• Stokes Equation
• High Order
• Finite Elements
• Saddle Point
• Systems
• Parallel
• Mpi
• High Performance Computing

Statement dc:rights

• Copyright 2024 Alexey Voronin

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/124245

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F124245