We developed a finite volume package FVM and a solver HYMLS, both based on elements of the Trilinos EPETRA-package (see http://trilinos.sandia.gov/). HYMLS is a linear system solver for steady state incompressible Navier-Stokes equations coupled to transport equations in 2 and 3D [1, 2, 3]. We constructed recently a multilevel variant of it, which makes it possible to solve 3D problems of over 10 million unknowns quickly on a parallel computer. The behavior of the method is very much like that of multigrid methods. In fact one could see it as the father of the multigrid method. The solver is very robust. For the problem described in [4], it allowed a quick increase in the Reynolds number to get into the interesting region around Re=2000. Here we will show the performance of the method on the Rayleigh-Bénard convection in a cube, with six no-slip walls [5]. Also here we employ HYMLS to solve the linear systems resulting from a Cayley transform of the generalized eigenvalue problem.@en

We perform a numerical study of a two-component reaction–diffusion model. By using numerical continuation methods, combined with state-of-the-art sparse linear and eigenvalue solvers, we systematically compute steady state solutions and analyze their stability and relations in both two and three space dimensions. The approach gives a more reliable and complete picture than previous efforts based on time integration schemes and is also typically much more efficient in terms of computing time. We are therefore able to produce a rich bifurcation diagram showing a variety of solution patterns and transitions.@en