Section: Research Program

Multiphysics coupling and domain decomposition

Within our project, we start from the conception and analysis of models based on partial differential equations. Already at the PDE level, we address the question of coupling of different models; examples may be that of simultaneous fluid flow in a discrete network of two-dimensional fractures and in the surrounding three-dimensional porous medium, or that of interaction of a compressible flow with the surrounding elastic deformable structure. The key physical characteristics need to be captured, whereas existence, uniqueness, and continuous dependence on the data are minimal analytic requirements that we request to satisfy. At the modeling stage, we also plan to develop model-order reduction techniques, such as the use of reduced basis techniques or proper generalized decompositions, to tackle evolutive problems, in particular in the nonlinear case.

We also concentrate an important effort on the development and analysis of efficient solvers for the systems of nonlinear algebraic equations. We have already in the past developed Newton–Krylov solvers, with a particular impact on parallelization that we achieve via domain decomposition. Here we specialize on Robin boundary conditions, where an optimized choice of the parameter has already shown speed-ups in orders of magnitude in terms of the number of the iterations of the domain decomposition algorithm in question. A novel feature is the use of such algorithms in time-dependent problems in space-time domain decomposition which allows the use of different time steps in different parts of the computational domain. This is particularly useful in porous media applications, where the amount of diffusion (permeability) varies abruptly, so that the evolution speeds vary importantly and call for adapted localized time stepping. Our other novel theme are Newton–multigrid solvers, where the geometric multigrid solver ingredients are tailored to the specific problem under consideration and to the specific numerical method, with problem- and discretization-dependent restriction, prolongation, and smoothing. This in particular yields mass balance on each iteration step, a very welcome feature in most of the target applications. The solver itself is then steered adaptively at each execution step by an a posteriori error estimate.