A saturation property for the spectral-Galerkin approximation of a Dirichlet problem in a square

TL;DR

This paper investigates how increasing the polynomial degree in spectral-Galerkin methods affects error reduction for Dirichlet problems, establishing that increments proportional to the degree p ensure p-independent error decrease.

Contribution

It demonstrates that increasing the polynomial degree by an amount proportional to p guarantees p-robust error reduction in spectral-Galerkin approximations for the Poisson problem.

Findings

Increment proportional to p yields p-robust error reduction.

Constant increment does not guarantee p-independent error decrease.

Provides computational evidence supporting theoretical results.

Abstract

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.