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

¹ Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
² Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD, USA
³ Korteweg-de Vries Institute for Mathematics, University of Amsterdam, PO Box 94248, 1090 GE Amsterdam, The Netherlands
⁴ MOX-Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, I-20133 Milano, Italy

^* Corresponding author: claudio.canuto@polito.it

Received: 20 December 2017
Accepted: 18 February 2019

Abstract

Both practice and analysis of p-FEMs and adaptive 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.