## Section: Research Program

### Domain decomposition and Newton–Krylov (multigrid) solvers

We next concentrate an intensive effort on the development and analysis of
efficient solvers for the systems of nonlinear algebraic equations that
result from the above discretizations. We have in the past developed *Newton–Krylov solvers* like the *adaptive inexact Newton method*, and we place
a particular emphasis on *parallelization* achieved via the *domain
decomposition* method. Here we traditionally specialize in *Robin
transmission conditions*, where an optimized choice of the parameter has
already shown speed-ups in orders of magnitude in terms of the number of
domain decomposition iterations in model cases. We concentrate in the SERENA
project on adaptation of these algorithms to the above novel discretization
schemes, on the optimization of the free Robin parameter for challenging
situations, and also on the use of the Ventcell transmission conditions.
Another feature is the use of such algorithms in time-dependent problems in
*space-time* domain decomposition that we have recently pioneered. This
allows the use of different time steps in different parts of the
computational domain and turns out to be particularly useful in porous media
applications, where the amount of diffusion (permeability) varies abruptly,
so that the evolution speed varies significantly from one part of the
computational domain to another. Our new theme here are *Newton–multigrid solvers*, where the geometric multigrid solver is *tailored* to the specific problem under consideration and to the specific
numerical method, with problem- and discretization-dependent restriction,
prolongation, and smoothing. Using patchwise smoothing, we have in particular recently developed a first multigrid method whose behavior is both in theory and in practice insensitive of (robust with respect to) the approximation polynomial degree. With patchwise techniques, we also achieve mass balance at each iteration step, a highly demanded feature in most of the target applications.
The solver itself is then *adaptively steered* at each execution step by
an a posteriori error estimate (adaptive stepsize, adaptive smoothing).