Contraction and optimality properties of adaptive Legendre-Galerkin methods: the 1-dimensional case

TL;DR

This paper analyzes the convergence, contraction, and optimality of adaptive Legendre-Galerkin methods for 1D elliptic boundary-value problems, demonstrating their effectiveness and optimality in approximating solutions with certain regularity.

Contribution

It provides a rigorous analysis of adaptive Legendre-Galerkin methods, including convergence, enhanced performance strategies, and proof of optimality within Gevrey-type sparsity classes.

Findings

Proves convergence at a fixed rate for an ideal adaptive algorithm.

Enhances performance by activating more degrees of freedom per iteration.

Guarantees optimality through a coarsening step, achieving near-best approximation.

Abstract

As a first step towards a mathematically rigorous understanding of adaptive spectral/$hp$ discretizations of elliptic boundary-value problems, we study the performance of adaptive Legendre-Galerkin methods in one space dimension. These methods offer unlimited approximation power only restricted by solution and data regularity. Our investigation is inspired by a similar study that we recently carried out for Fourier-Galerkin methods in a periodic box. We first consider an "ideal" algorithm, which we prove to be convergent at a fixed rate. Next we enhance its performance, consistently with the expected fast error decay of high-order methods, by activating a larger set of degrees of freedom at each iteration. We guarantee optimality (in the non-linear approximation sense) by incorporating a coarsening step. Optimality is measured in terms of certain sparsity classes of the Gevrey type, which describe a (sub-)exponential decay of the best approximation error.