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A parallel Newton multigrid framework for monolithic fluid-structure interactions

We present a monolithic parallel Newton-multigrid solver for nonlinear three dimensional fluid-structure interactions in Arbitrary Lagrangian Eulerian formulation. We start with a finite element discretization of the coupled problem, based on a remapping of the Navier-Stokes equation onto a fixed reference framework. The strongly coupled fluid-structure interaction problem is discretized with finite elements in time and finite differences in time. The resulting nonlinear and linear systems of equations are large and show a very high condition number. We present a novel Newton approach that is based on two essential ideas: First, a static condensation of solid deformation by exploiting the velocity-deformation relation $d_t u = v$. Second, the Jacobian of the fluid-structure interaction system is simplified by neglecting all derivatives with respect to the ALE deformation, an approximation that has shown to have little impact. The resulting system of equation decouples into a joint momentum equation and into two separated equations for the deformation fields in solid and fluid. Besides a reduction of the problem sizes, the approximation has a positive effect on the conditioning of the systems such that multigrid solvers with simple smoothers like a parallel Vanka-iteration can be applied. We demonstrate the efficiency of the resulting solver infrastructure on a well-studied 2d test-case and we also introduce a challenging 3d problem. For 3d problems we achieve a substantial acceleration as compared to established approaches found in literature.