An adaptive hp-version of the finite element method applied to flame propagation problems

This paper describes an adaptive hp-version mesh re nement strategy and its application to the nite element solution of one-dimensional ame propagation problems. The aim is to control the spatial and time discretiza- tion errors below a prescribed error tolerance at all time levels. In the algorithm, the optimal time step is rst determined in an adaptive manner by considering the variation of the computable error in the reaction zone. Later, the method uses a p-version re nement till the computable a posteriori error is brought down below the tolerance. During the p-version, if the maximum allowable degree of approximation is reached in some elements of the mesh without satisfying the global error tolerance criterion, then conversion from p- to h-version is performed. In the conversion procedure, a gradient based non-uniform h-version re nement has been introduced in the elements of higher degree approximation. In this way, p-version and h-version approaches are used alternately till the a posteriori error criteria are satis ed. The mesh re nement is based on the element error indicators, according to a statistical error equi-distribution procedure. Numerical simu- lations have been carried out for a linear parabolic problem and premixed ame propagation in one-space dimension