Fast and Reliable Solution of the Navier - Stokes EquationsIncluding

In this paper, we describe recent developments in the design and implementation of Navier-Stokes solvers based on nite element discretization. The most important ingredients are residual driven a posteriori mesh reenement, fully coupled defect-correction iteration for linearization, and optimal multigrid precondi-tioning. These techniques were systematically developed for computing incompressible viscous ows in general domains. Recently they have been extended to compress-ible low-Mach ows involving chemical reactions. The potential of automatic mesh adaptation together with multilevel techniques is illustrated by several examples, (1) the accurate prediction of drag and lift coeecients, (2) the determination of CARS-signals of species concentration in ow reactors, (3) the computation of laminar ames. 1 The principle of error estimation First, we introduce the concept underlying our approach to error estimation and mesh optimization. The goal is to develop techniques for (i) reliable estimation of the discretization error in quantities of physical interest, (ii) eeective and economical mesh adaptation, and (iii) fast solution of the discretized problems by multi-level techniques. The use of nite element Galerkin discretization provides the appropriate framework for a mathematically rigorous a posteriori error analysis. 1.1 The Discretization Error and its Cause The total discretization error in a mesh cell T splits into two components, the locally produced \truncation error" and the globally transported \pollution error": E tot T = E loc T + E trans T : In a Galerkin method the eeect of the residual T , in a cell T , on the local error E T 0 , at another cell T 0 , is governed by generalized Green's functions of the continuous problem (see Figure 1). In the same way, the dependence of the local error on the various physical mechanisms inherent to the problem to be solved can be described. This is the general concept underlying our approach to error control. The crucial questions for an eeective control of the error are: { How can we detect and use the interplay of the various error propagation eeects (local diiusion, directional uid transport, incompressibility constraint, stii reaction terms, etc.) for the design of economical meshes in solving coupled problems: Given N cells, what is the \best" distribution of grid cells? { How can we achieve accurate a posteriori error control for quantities of physical interest (e.g., drag and lift in ows around bodies, point values of the temperature in ames, mean values of species-concentrations in ow reactors, etc.)? …