Two-level additive Schwarz preconditioners with GDSW coarse spaces for high-order discretizations in 2D

Abstract

Two-level overlapping Schwarz preconditioners with Generalized Dryja-Smith-Widlund (GDSW) and Reduced GDSW (RGDSW) coarse spaces for high-order finite element methods (FEM) and virtual element methods (VEM) are investigated and applied to the Poisson problem in 2D. The GDSW coarse space is a well-established robust choice for preconditioners in domain decomposition, while the RGDSW coarse space offers a competitive alternative by reducing the coarse problem dimension while maintaining robustness. On the theoretical side, a condition number bound for both preconditioners is proved for conforming $\mathbb{P}_k$ and $\mathbb{Q}_k$ finite element spaces with explicit dependence on the polynomial degree $k$. Through numerical experiments on structured triangular and quadrilateral meshes for FEM and on polygonal meshes for VEM, it is demonstrated that both GDSW and RGDSW maintain scalability properties across polynomial orders up to $k=9$ for selected degrees. The RGDSW approach achieves a reduction in the coarse problem dimension by a factor of approximately three compared to GDSW while maintaining competitive iteration counts.