An Adaptive Finite Element Method

We develop an adaptive nite element algorithm for reactive ow simulations. The underlying equations are the Navier-Stokes equations for low-Mach-number ows and an additional system of convection-diiusion-reaction equations for the chemical species. Especially in combustion problems the reaction terms are very stii and nonlinear. This set of equations represents a complicated system of PDEs which is solved usually on parallel computers due to the high numerical cost of memory and computing time. Adaptive methods lead to an important reduction of the computational cost for this type of equations if a good error estimator is available. Our adaptive process is based on an a posteriori error estimate. We use duality arguments in order to control the error in functionals of the solution like, e.g., line-averaged radical concentrations in a stationary ame front. By solving a linearized dual problem, we determine weights multiplying the local residuals in the error estimate. This provides information about the global error propagation, which is used in a criterion for local mesh adaptation.