Multigrid and domain decomposition methods provide efficient algorithms for the numerical solution of partial differential equations arising in the modeling of many applications in Computational Science and Engineering. This manuscript covers certain aspects of modern iterative solution methods for the solution of large-scale problems issued from the discretization of partial differential equations. More specifically, we focus on geometric multigrid methods, non-overlapping substructuring methods and flexible Krylov subspace methods with a particular emphasis on their combination. First, the combination of multigrid and Krylov subspace methods is investigated on a linear partial differential equation modeling wave propagation in heterogeneous media. Secondly, we focus on non-overlapping domain decomposition methods for a specific finite element discretization known as the $hp$ finite element, where unrefinement/refinement is allowed both by decreasing/increasing the step size h or by decreasing/increasing the polynomial degree p of the approximation on each element. Results on condition number bounds for the domain decomposition preconditioned operators are given and illustrated by numerical results on academic problems in two and three dimensions. Thirdly, we review recent advances related to a class of Krylov subspace methods allowing variable preconditioning. We detail flexible Krylov subspace methods including augmentation and/or spectral deflation, where deflation aims at capturing approximate invariant subspace information. We also present flexible Krylov subspace methods for the solution of linear systems with multiple right-hand sides given simultaneously. The efficiency of the numerical methods is demonstrated on challenging applications in seismics requiring the solution of huge linear systems of equations with multiple right-hand sides on parallel distributed memory computers. Finally, we expose current and future prospectives towards the design of efficient algorithms on extreme scale machines for the solution of problems issued from the discretization of partial differential equations.
Iain Duff – Examiner (Rutherford Appleton Laboratory, UK and Cerfacs, France)
Andreas Frommer – Referee (University of Wuppertal, Germany)
Serge Gratton – Examiner (INPT-IRIT, France)
Frédéric Nataf – Referee (Université Pierre et Marie Curie, France)
Cornelis Oosterlee – Referee (CWI and TU Delft, The Netherlands)
Michel Visonneau – Examiner (Ecole Centrale de Nantes, France)