  This work surveys an r-adaptive moving mesh finite element method for the
numerical solution of premixed laminar flame problems. Since the model of
chemically reacting flow involves many different modes with diverse length
scales, the computation of such a problem is often extremely time-consuming.
Importantly, to capture the significant characteristics of the flame structure
when using detailed chemistry, a much more stringent requirement on the spatial
resolution of the interior layers of some intermediate species is necessary.
Here, we propose a moving mesh method in which the mesh is obtained from the
solution of so-called moving mesh partial differential equations. Such
equations result from the variational formulation of a minimization problem for
a given target functional that characterizes the inherent difficulty in the
numerical approximation of the underlying physical equations. Adaptive mesh
movement has emerged as an area of intense research in mesh adaptation in the
last decade. With this approach points are only allowed to be shifted in space
leaving the topology of the grid unchanged. In contrast to methods with local
refinement, data structure hence is unchanged and load balancing is not an
issue as grid points remain on the processor where they are. We will
demonstrate the high potential of moving mesh methods for effectively
optimizing the distribution of grid points to reach the required resolution for
chemically reacting flows with extremely thin boundary layers.
