1 preprint numerics no. 1/2012 norwegian university of science and technology trondheim, norway This paper discusses numerical solution of boundary value problems using spectral methods combined with nonlinear and adaptive mappings between the reference domain and the physical domain. A brief review of existing methods for adaptive mesh generation is given, and a method for finding close to opt...

In the numerical solution of practical problems of physics or engineering such as, e.g., computational fluid dynamics, elasticity, or semiconductor device simulation one often encounters the difficulty that the overall accuracy of the numerical approximation is deteriorated by local singularities such as, e.g., singularities arising from re-entrant corners , interior or boundary layers, or shar...

Adaptive methods are now widely used in the scientific computation to achieve better accuracy with minimum degree of freedom. In this chapter, we shall briefly survey recent progress on the convergence analysis of adaptive finite element methods (AFEMs) for second order elliptic partial differential equations and refer to Nochetto, Siebert and Veeser [14] for a detailed introduction to the theo...

We study the performance of adaptive Fourier-Galerkin methods in a periodic box in R d with dimension d ≥ 1. These methods offer unlimited approximation power only restricted by solution and data regularity. They are of intrinsic interest but are also a first step towards understanding adaptivity for the hp-FEM. We examine two nonlinear approximation classes, one classical corresponding to alge...