An algebraic 2-level domain decomposition preconditioner with applications to the compressible Euler equations

A domain decomposition technique to solve large-scale aerodynamics problems on unstructured grids is investigated. The linear system, arising when an implicit time-advancing scheme is used, is preconditioned using a Schwarz-based method. The key idea of the Schwarz preconditioner is to solve (approximately) a linear problem on each subdomain, then to exchange information only with neighbouring subdomains. Unfortunately, the performance of the Schwarz-type preconditioner deteriorates as the number of processors grows. So, in this case, a key element for obtaining a scalable preconditioner is to provide a coarse level operator. Since many of the coarse operators proposed in literature are difficult to implement on unstructured 2D and 3D meshes, a purely algebraic procedure, that requires the entries of the matrix only, has been developed. This procedure may be seen as an Algebraic MultiGrid (AMG) method applied as a coarse grid correction operator. The key idea is to take advantage of the use of local data of domain decomposition preconditioners, and of the automatic coarsening procedures of AMG methods. Two possible schemes to introduce the coarse grid operator will be described. Both cases have been implemented and tested in a distributed parallel environment, using the MPI library. It will be shown that for suitable values of the rank of the coarse grid operator it is possible to obtain a considerable reduction in the number of iterations compared to the Schwarz preconditioner without coarse operato