Generalized sweeping preconditioners for domain decomposition methods applied to Helmholtz problems

The numerical solution of high-frequency Helmholtz problems by discretization methods such as the finite element method is a big challenge. Indeed, obtaining high-fidelity solutions requires to assemble and solve extremely large linear systems, whose size increases more than linearly with frequency. This can quickly lead to intractable computational costs both in terms of the assembly and the solution of the resulting linear systems. To overcome these difficulties, we implement an efficient quadrature approach applied to the high-order finite element method as well as domain decomposition methods with high-order transmission conditions, in two- and three dimensions, on high-performance computers. To improve the convergence rate of domain decomposition methods, we generalize a family of sweeping preconditioners for the domain decomposition methods, where sweeps can be done in several directions on block-type partitions. In order to apply our algorithms to practical cases that require solutions for large number of frequencies or a large number of right-hand sides, we also propose improved parallelization strategies that maintain the fast rate of convergence while maximizing the usage of computer resources.(FSA - Sciences de l'ingénieur) -- UCL, 202