Adaptive Spectral Element Methods for Turbulence and Transition

These notes present an introduction to the spectral element method with applications to fluid dynamics. The method is introduced for one-dimensional problems, followed by the discretization of the advection and diffusion operators in multi-dimensions, and efficient ways of dealing with these operators numerically. We also discuss the mortar element method, a technique for incorporating local mesh refinement using nonconforming elements; this is the foundation for adaptive methods. An adaptive strategy based on analyzing the local polynomial spectrum is presented and shown to give accurate solutions even for problems with weak singularities. Finally we describe techniques for integrating the incompressible Navier-Stokes equations, including methods for performing computational linear and nonlinear stability analysis of non-parallel and time-periodic flows.