Nonlinear Preconditioning: How to use a Nonlinear Schwarz Method to Precondition Newton's Method

For linear problems, domain decomposition methods can be used directly as iterative solvers, but also as preconditioners for Krylov methods. In practice, Krylov acceleration is almost always used, since the Krylov method finds a much better residual polynomial than the stationary iteration, and thus converges much faster. We show in this paper that also for non-linear problems, domain decomposition methods can either be used directly as iterative solvers, or one can use them as preconditioners for Newton's method. For the concrete case of the parallel Schwarz method, we show that we obtain a preconditioner we call RASPEN (Restricted Additive Schwarz Preconditioned Exact Newton) which is similar to ASPIN (Additive Schwarz Preconditioned Inexact Newton), but with all components directly defined by the iterative method. This has the advantage that RASPEN already converges when used as an iterative solver, in contrast to ASPIN, and we thus get a substantially better preconditioner for Newton's method. The iterative construction also allows us to naturally define a coarse correction using the multigrid full approximation scheme, which leads to a convergent two level non-linear iterative domain decomposition method and a two level RASPEN non-linear preconditioner. We illustrate our findings with numerical results on the Forchheimer equation and a non-linear diffusion problem

An unconditionally stable staggered pressure correction scheme for the compressible Navier-Stokes equations

International audienceIn this paper we present a pressure correction scheme for the compressible Navier-Stokes equations. The space discretization is staggered, using either the Marker-And Cell (MAC) scheme for structured grids, or a nonconforming low-order finite element approximation for general quandrangular, hexahedral or simplicial meshes. For the energy balance equation, the scheme uses a discrete form of the conservation of the internal energy, which ensures that this latter variable remains positive; this relation includes a numerical corrective term, to allow the scheme to compute correct shock solution in the Euler limit. The scheme is shown to have at least one solution, and to preserve the stability properties of the continuous problem, irrespectively of the space and time steps. In addition, it naturally boils down to a usual projection scheme in the limit of vanishing Mach numbers. Numerical tests confirm its potentialities, both in the viscous incompressible and Euler limits

Model reduction for coupled near-well and reservoir models using multiple space-time discretizations

In reservoir simulations, fine fully-resolved grids deliver accurate model representations, but lead to large systems of nonlinear equations to solve every time step. Numerous techniques are applied in porous media flow simulations to reduce the computational effort associated with solving the underlying coupled nonlinear partial differential equations. Many models treat the reservoir as a whole. In other cases, the near-well accuracy is important as it controls the production rate. Near-well modeling requires finer space and time resolution compared with the remaining of the reservoir domain. To address these needs, we combine Model Order Reduction (MOR) with local grid refinement and local time stepping for reservoir simulations in highly heterogeneous porous media. We present a domain decomposition algorithm for a gas flow model in porous media coupling near-well regions, which are locally well-resolved in space and time with a coarser reservoir discretization. We use a full resolution for the near-well regions and apply MOR in the remainder of the domain. We illustrate our findings with numerical results on a gas flow model through porous media in a heterogeneous reservoir