Effects of integrations and adaptivity for the Eulerian-Lagrangian method

This paper provides an analysis on the effects of exact and inexact integrations on stability, convergence, numerical diffusion, and numerical oscillations for the Eulerian-Lagrangian method (ELM). In the finite element ELM, when more accurate integrations are used for the right-hand-side, less numerical diffusion is introduced and better approximation is obtained. When linear interpolation is used for numerical integrations, the resulting ELM is shown to be unconditionally stable and of first-order accuracy. When Gauss quadrature is used, conditional stability and second-order accuracy are established under some mild constraints for the convection-diffusion problems. Finally, numerical experiments demonstrate that more accurate integrations lead to better approximation, and spatial adaptivity can substantially reduce numerical oscillations and smearing that often occur in the ELM when inexact numerical integrations are used. Copyright 2011 by AMSS, Chinese Academy of Sciences.