Domain decomposition preconditioners

The purpose of this thesis is to define efficient parallel preconditioners based on the domain decomposition paradigm and to apply them to the solution of the steady compressible Euler equations. In the first part we propose and analyse various domain decomposition preconditioners of both overlapping (Schwarz) and non-overlapping (Schur complement-based) type. For the former, we deal with two-level methods, with an algebraic formulation of the coarse space. This approach enjoys several interesting properties not always shared by more standard twolevel methods. For the latter, we introduce a class of preconditioners based on a peculiar decomposition of the computational domain. The domain is decomposed in such a way that one subdomain is connected to all the others, which are in fact disconnected components. A class of approximate Schur complement preconditioners is also considered. Theoretical and numerical results are given for a model problem. In the second part we consider the application of the previous domain decomposition preconditioners to the compressible Euler equations. The discretisation procedure, based on multidimensional upwind residual distribution schemes, is outlined. We introduce a framework that combines non-linear system solvers, Krylov accelerators, domain decomposition preconditioners, as well as mesh adaptivity procedures. Several numerical tests of aeronautical interest are carried out in order to assess both the discretisation schemes and the mesh adaptivity procedures. In the third part we consider the parallel aspects inherent in the numerical solution of the compressible Euler equations on parallel computers with distributed memory. All the main kernels of the solution algorithm are analysed. Many numerical tests are carried out, with the aim of investigating the performance of the domain decomposition preconditioners proposed in the first part of the thesis, in the applications addressed in the second part.