Contraction and optimality properties of adaptive Legendre–Galerkin methods: The one-dimensional case

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.