Fluid-structure interaction problems arise in many fields of application such as flows around elastic structures or blood flow in arteries. The method presented in this paper for solving such a problem is based on a reduction to an equation at the interface, involving the so-called Steklov-Poincaré operators. This interface equation is solved by a Newton iteration, for which directional derivatives involving shape derivatives with respect to the interface perturbation have to be evaluated appropriately. One step of the Newton iteration requires the solution of several decoupled linear sub-problems in the structure and the fluid domains. These sub-problems are spatially discretized by a finite element method on hybrid meshes. For the time discretization implicit first-order methods are used for both sub-problems. The discretized equations are solved by algebraic multigrid methods.
In this paper, we consider a monolithic approach to handle coupled fluid-structure interaction problems with different hyperelastic models in an all-at-once manner. We apply Newtons method in the outer iteration dealing with nonlinearities of the coupled system. We discuss preconditioned Krylov sub-space, algebraic multigrid and algebraic multilevel methods for solving the linearized algebraic equations. Finally, we compare the results of the monolithic approach with those of the corresponding partitioned approach that was studied in our previous work.
Summary
The aim of this work is to compare algebraic multigrid (AMG) preconditioned GMRES methods for solving the nonsymmetric and positive definite linear systems of algebraic equations that arise from a space–time finite‐element discretization of the heat equation in 3D and 4D space–time domains. The finite‐element discretization is based on a Galerkin–Petrov variational formulation employing piecewise linear finite elements simultaneously in space and time. We focus on a performance comparison of conventional and modern AMG methods for such finite‐element equations, as well as robustness with respect to the mesh discretization and the heat capacity constant. We discuss different coarsening and interpolation strategies in the AMG methods for coarse‐grid selection and coarse‐grid matrix construction. Further, we compare AMG performance for the space–time finite‐element discretization on both uniform and adaptive meshes consisting of tetrahedra and pentachora in 3D and 4D, respectively. The mesh adaptivity occurring in space and time is guided by a residual‐based a posteriori error estimation.
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