Two-level additive Schwarz preconditioners with GDSW coarse spaces for high-order discretizations in 3D

Overlapping Schwarz methods are among the most successful domain decomposition preconditioners for large-scale finite element simulations. In this work, the Generalized Dryja-Smith-Widlund (GDSW), GDSW$^*$, and Reduced GDSW (RGDSW) coarse spaces are studied for three-dimensional high-order finite element method (FEM) and virtual element method (VEM) for the Poisson equation and the equations of linear elasticity. To the best of our knowledge, this is the first study of GDSW-type coarse spaces for polynomial degrees $k\ge 3$ in three dimensions, for either discretization. In 3D VEM, the face degrees of freedom (DOFs) are moments instead of point values. The kernel of the differential operator, which defines the interface values of the GDSW coarse basis functions, must be represented through these moments. The construction is therefore no longer purely algebraic and, without this correction, the coarse level loses its effect for $k\ge 3$. FEM and VEM are tested on hexahedral meshes, and VEM is additionally tested on polyhedral ones. Polynomial degrees up to $k=5$ and runs on up to $4\,096$ subdomains and parallel cores are included in the numerical study. Numerical experiments show scalable iteration counts for all coarse space variants for the Poisson problem, illustrating the expected trade-off between the coarse space size and robustness. Further experiments for linear elasticity confirms that the moment-based construction extends naturally from the scalar to the vector-valued problem.