Parallel Two-Level Methods for Three-Dimensional Transonic Compressible Flow Simulations on Unstructured Meshes*

It is well known, at least in theory, that the coarse space plays a very important role in the fast and scalable convergence of the algorithms. Direct methods are often used to solve the coarse mesh problem either redundantly on all processors or on a subset of processors. This presents a major difficulty in a parallel implementation for 3D problems, especially when the number of processors is large. The chapter proposes several techniques for solving the coarse mesh problem in parallel, together with the local fine mesh problems, using two nested layers of preconditioned iterative methods. The construction of the coarse mesh is an interesting issue by itself. A different approach is taken from what is commonly used in the algebraic multigrid methods in which the coarse mesh is obtained from the given fine mesh—not the given geometry. In the two level methods presented in the chapter, both the coarse mesh and the fine mesh are constructed from the given geometry. To better fit the boundary geometry, the fine mesh nodes may not be on the faces of the coarse mesh tetrahedrons. In other words, the coarse space and the fine space are not nested. This does not present a problem as long as the proper interpolation is defined.