Thèse de doctorat
Résumé : We investigate the numerical solution of saddle-point linear systems arising from the discretizationof Stokes and Oseen problems. We consider all-at-once algebraic multigrid (AMG) methods whichtypically struggle when applied to such systems. To avoid the usual difficulties, we introduce anew approach called the transform-then-solve approach. The key idea is to first apply an algebraictransformation to the linear system in order to give it a more suitable structure for the subsequentapplication of a relatively standard aggregation-based AMG method.The investigation of typical (low-order) finite elements discretization of Stokes problems show thatthe method is robust on a broad range of problems, even for variable viscosity or time-dependentproblems used in an industrial framework. The main challenge consists in improving the compet-itiveness of the method with respect to reference methods, because of the higher computationalcost of the method due to the algebraic transformation. To mitigate it, we implement two strate-gies which efficiently reduce the computational cost. As a consequence, the transform-then-solveapproach is competitive and significantly more robust.The method developed during the thesis is sequential by design. However, the use of parallelarchitecture has been shown to speed up significantly computations. As such, we also look at thealgorithmic changes needed to develop a parallel version of the transform-then-solve approach.We show that these changes do not impact the robustness of the method.The development of an AMG method for Oseen problems is more complicated because of thenon-symmetry of the velocity block. By using the convergence theory we develop in the thesis,we are able to show that it is possible to design a two-grid method which presents a convergenceindependent of the mesh size and the Reynolds number for constant convection problem. The keycondition is that the aggregation of the pressure unknowns follows the same pattern as the one ofthe velocity unknowns. This robust convergence is observed to hold for recirculating flows.However, it is difficult to transpose these results in a multigrid setting. Indeed, the aggregationof the pressure leads to a modification of the structure of the pressure block at coarser levels. Asa consequence, the convergence breaks down when going from two- to multigrid. Improving thesmoother by an AMG approximation of the pressure block allows to retrieve the robustness in theReynolds number. Unfortunately, the robustness in the mesh size is lost and the resulting methodis not optimal.