On non-overlapping domain decomposition preconditioners for discontinuous Galerkin finite element methods in $H^{2}$-type norms - Smears, I (Unspecified, 2013)
We analyse the spectral bounds of non-overlapping domain decomposition additive Schwarz preconditioners for $hp$-version discontinuous Galerkin finite element methods in $H^{2}$-type norms. Using original approximation results for discontinuous finite element spaces, it is found that these preconditioners yield a condition number bound of order $1 + H^{3}p^{6}/h^{3}q^{3}$, where $H$ and $h$ are respectively the coarse and fine mesh sizes, and $q$ and $p$ are respectively the coarse and fine mesh polynomial degrees. Numerical experiments show that the orders of the spectral bounds are sharp.