Adaptive spectral methods

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 optimal mappings for boundary value problems in R 1 is presented. The method exploits the link between high order numerical solutions of PDEs and approximation of parametric curves. Also, other adaptive methods for boundary value problems in R 1 are proposed, based either on minimizing the discrete L 2-norm of the residual, or interpolating the residual as a parametric curve. The adaptive methods are constructed with the aim of finding optimal mappings, however, this turns out to be a very difficult task. Still, significant improvement from standard (non-adaptive) high order methods is achieved in some cases.