Adaptive Finite Element Methods for Low-mach-number Flows with Chemical Reactions

Introduction We describe recent developments in the design and implementation of nite element methods for the Navier-Stokes equations modelling chemically reactive ows. The emphasize is on the low-Mach number regime including the limit case of incompressible ow. The most important ingredients are residual driven a posteriori mesh reenement, fully coupled defect-correction iteration for linearization, and optimal multigrid precondition-ing. These computational techniques are systematically developed for several linearized sub-problems and are then applied to the full model of compressible low-Mach number ow including chemical reactions. The potential of automatic mesh adaptation together with multilevel techniques is illustrated by various examples. In Chapters 1, we discuss the governing equations of compressible ow in the low-Mach-number regime and the particularities introduced by the inclusion of chemical reactions. Then, Chapter 2 presents our nite element discretization of the resulting models with emphasize on the description of pressure stabilization and numerical diiusion by least-squares techniques. Chapter 3 introduces into a new approach to error control and adaptive mesh design which is based on weighted a posteriori error estimates derived by duality techniques. In Chapter 4, the solution process involving defect-correction and multigrid techniques is described. Chapter 5 presents some applications of our methods to solving compressible ow problems including chemical reaction systems of diierent complexity. In the Appendix, we collect some details about the models of chemical reactions used in our simulations.