A saturation property for the spectral-Galerkin approximation of a Dirichlet problem in a square
Wednesday 3rd January 2018
Canuto, C.; Nochetto, R. H.; Stevenson, R.; Verani, M.
Both practice and analysis of adaptive $p$-FEMs and $hp$-FEMs raise the question what increment in the current polynomial degree $p$ guarantees a $p$-independent reduction of the Galerkin error. We answer this question for the $p$-FEM in the simplified context of homogeneous Dirichlet problems for the Poisson equation in the two dimensional unit square with polynomial data of degree $p$. We show that an increment proportional to $p$ yields a $p$-robust error reduction and provide computational evidence that a constant increment does not.