Mini - symposium A 14 High - Order Methods and Their Applications

We design adaptive high-order Galerkin methods for the solution of linear elliptic problems and study their performance. We start by considering adaptive Fourier-Galerkin methods and Legendre spectralelement methods, which offer unlimited approximation power only restricted by solution and data regularity. We next turn our attention to the $hp$-version of the finite element method, for which we design an adaptive scheme which hinges on a recent algorithm by P. Binev for adaptive $hp$-approximation. We prove convergence with contraction rate, and we investigate the optimality properties of the algorithm.