Two New Domain Decomposition Solvers

Einar M. Ronquist (Nektonics, Inc.)

We present two new domain decomposition solvers in the context of spectral element discretizations. The first is a domain decomposition solver for the discrete steady convection-diffusion equation, while the second is a domain decomposition solver for the discrete steady Stokes or Navier-Stokes equations. The solution algorithms are both based on the additive Schwarz method in the context of nonoverlapping subdomains. The key ingredients are: (i) a coarse global system; (ii) a set of local, independent subproblems associated with the subdomains (or spectral elements); (iii) a system associated with the unknowns on the subdomain interfaces; and (iv) a Krylov method such as the CG algorithm or the GMRES algorithm. We present numerical results that demonstrate the convergence properties of the new solvers, as well as the applicability of the methods to solve heat transfer and incompressible fluid flow problems.