A saturation property for the spectral-Galerkin approximation of a Dirichlet problem in a square
This addresses a fundamental question for adaptive p- and hp-FEM practitioners, providing theoretical and numerical guidance on degree increments.
The paper investigates the increment in polynomial degree needed for p-robust error reduction in p-FEM for Poisson problems, showing that a proportional increment works while a constant one does not.
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.