Adaptive control of error and stability of h-p approximations of the transient Navier-Stokes equations

A posteriori error estimation in finite element analysis has become an essential tool to develop reliable adaptive methods which improve the efficiency and accuracy of numerical simulations. In this dissertation, a new methodology, rigorous mathematically and easily applicable, is proposed to estimate and control the numerical errors in finite element approximations of the time-dependent incompressible Navier-Stokes equations. The approach is based on the definition of two residuals, namely the residual with respect to the momentum equation and the residual with respect to the continuity equation. These residuals, which represent the sources of errors due to the finite element discretization, are shown to provide two error estimates in specific norms of the solution space. These estimates are local in time and do not reflect the errors due to the time discretization.; The error estimates are utilized to design an adaptive strategy, in which the finite element mesh is automatically adjusted when and where it is necessary during the flow evolution. A clear advantage in the present approach is to treat the residuals individually, as a study of the numerical stability with respect to the mesh size reveals that the residual in the continuity equation is responsible for unphysical and unstable perturbations. Thus, the proposed adaptation scheme takes in account this residual first when refining the mesh. The performance of this strategy for error and stability control is demonstrated on two-dimensional applications with moderate Reynolds numbers.; Another important collection of new results presented in this study deals with error estimates in quantities of interest, by which is meant quantities of the solution that can be characterized as linear functionals on the solution spaces, such as a component of the velocity or stress at a point. This theory represents a significant alternative to the existing theory of a posteriori error estimation which is primarily concerned with global, energy-norm estimates. Applications of the technique to an elliptic model problem and to the Stokes equations show that such estimates are quite accurate and that upper and lower bounds on the error can be attained. The concept of goal-oriented adaptivity is then introduced, which embodies adaptation procedures designed to control the approximation error in specific quantities of interest. Numerical experiments suggest that such procedures greatly accelerate the calculation of important features of the solution to the levels of accuracy as compared to traditional adaptive schemes based on global estimates.