Date of Award
Doctor of Philosophy (PhD)
Dr. Timo Heister, Committee Chair
Dr. Leo Rebholz
Dr. Christopher Cox
Dr. Qingshan Chen
Numerical solutions to fluid flow problems involve solving the linear systems arising from the discretization of the Stokes equation or a variant of it, which often have a saddle point structure and are difficult to solve. Geometric multigrid is a parallelizable method that can efficiently solve these linear systems especially for a large number of unknowns. We consider two approaches to solve these linear systems using geometric multigrid: First, we use a block preconditioner and apply geometric multigrid as in inner solver to the velocity block only. We develop deal.II tutorial step-56 to compare the use of geometric multigrid to other popular alternatives. This method is found to be competitive in serial computations in terms of performance and memory usage. Second, we design a special smoother to apply multigrid to the whole linear system. This smoother is analyzed as a Schwarz method using conforming and inf-sup stable discretization spaces. The resulting method is found to be competitive to a similar multigrid method using non-conforming finite elements that were studied by Kanschat and Mao. This approach has the potential to be superior to the first approach. Finally, expanding on the research done by Dannberg and Heister, we explore the analysis of a three-field Stokes formulation that is used to describe melt migration in the earth's mantle. Multiple discretizations were studied to find the best one to use in the ASPECT software package. We also explore improvements to ASPECT's linear solvers for this formulation utilizing block preconditioners and algebraic multigrid.
Grove, Ryan R., "Discretizations & Efficient Linear Solvers for Problems Related to Fluid Flow" (2017). All Dissertations. 1985.