Fast tensor product Schwarz smoothers for high-order discontinuous Galerkin methods - Witte, Julius; Arndt, Daniel; Kanschat, Guido (Computational methods in applied mathematics, 2021)
We discuss the efficient implementation of powerful domain decomposition smoothers for multigrid methods for high-order discontinuous Galerkin (DG) finite element methods. In particular, we study the inversion of matrices associated to mesh cells and to the patches around a vertex, respectively, in order to obtain fast local solvers for additive and multiplicative subspace correction methods. The effort of inverting local matrices for tensor product polynomials of degree k is reduced from 𝒪 ⁢ ( k 3 ⁢ d ) {\mathcal{O}(k^{3d})} to 𝒪 ⁢ ( d ⁢ k d + 1 ) {\mathcal{O}(dk^{d+1})} by exploiting the separability of the differential operator and resulting low rank representation of its inverse as a prototype for more general low rank representations in space dimension d .