High-Order Adaptive Galerkin Methods

We design adaptive high-order Galerkin methods for the solution of linear elliptic problems and study their performance. We first consider adaptive Fourier-Galerkin methods and Legendre-Galerkin methods, which offer unlimited approximation power only restricted by solution and data regularity. Their analysis of convergence and optimality properties reveals a sparsity degradation for Gevrey classes. We next turn our attention to the h p-version of the finite element method, design an adaptive scheme which hinges on a recent algorithm by P. Binev for adaptive h p-approximation, and discuss its optimality properties.