This is my keynote talk at the Leibniz MMS days 2026 in Frankfurt (Oder). In this presentation, we discuss recent progress and ongoing open questions in numerical methods for addressing coupled problems. Under the terminology coupled problems, we understand nonstationary, nonlinear, coupled PDE systems and variational inequalities (CVIS). Modeling includes space-time concepts. In Galerkin finite elements, care is taken for physics-based discretizations. Here, adaptivity on the discretization as well as solver level plays a crucial role to guarantee accuracies of certain goal functionals, while keeping the cost reasonable. Adaptivity includes (multi) goal-oriented error-control or predictor-corrector adaptivity or balancing iteration-discretization errors or adaptive non-intrusive global-local schemes. In addition, on the numerical solver level, we experimented over the last years with multigrid and domain decompositions preconditioners for iterative solvers such as CG or GMRES as well as model order reduction and the previously mentioned global-local schemes. Applications include incompressible flow (Navier-Stokes equations), fluid-structure interaction, phase-field fracture in porous media, and more recently time-harmonic Maxwell (the latter will not discussed due to time). Moreover, our progress in optimal control and parameter identification in fluid-structure interaction and phase-field fracture will be mentioned if time allows.
Thomas Wick (Fri,) studied this question.