Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?

The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed

Dual weighted residual based error control for nonstationary convection-dominated equations: potential or ballast?

Even though substantial progress has been made in the numerical approximation of convection-dominated problems, its major challenges remain in the scope of current research. In particular, parameter robust a posteriori error estimates for quantities of physical interest and adaptive mesh refinement strategies with proved convergence are still missing. Here, we study numerically the potential of the Dual Weighted Residual (DWR) approach applied to stabilized finite element methods to further enhance the quality of approximations. The impact of a strict application of the DWR methodology is particularly focused rather than the reduction of computational costs for solving the dual problem by interpolation or localization.Comment: arXiv admin note: text overlap with arXiv:1803.1064

A posteriori error estimation and adaptivity based on VMS for the incompressible Navier–Stokes equations

In this work an explicit a posteriori error estimator for the steady incompressible Navier–Stokes equations is investigated. The error estimator is based on the variational multiscale theory, where the numerical solution is decomposed in resolved scales (FEM solution) and unresolved scales (FEM error). The error is estimated locally considering the residuals that emerge from the numerical solution and the error inverse-velocity scales, t’s, associated with each type of residual. These error scales are provided in this paper, which have been computed a-priori solving a set of local problems with unit residuals. Therefore, the computational effort to predict the error is small and its implementation in any FEM code is simple. As an application, a strategy to develop adaptive meshes with the aim of optimizing the computational effort is shown. Numerical examples are presented to test the behavior of the error estimator

POD model order reduction with space-adapted snapshots for incompressible flows

We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two approaches of deriving stable POD-Galerkin reduced-order models for this context. In the first approach, the pressure term and the continuity equation are eliminated by imposing a weak incompressibility constraint with respect to a pressure reference space. In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the velocity reduced space with supremizers computed on a velocity reference space. For problems with inhomogeneous Dirichlet conditions, we show how suitable lifting functions can be obtained from standard adaptive finite element computations. We provide a numerical comparison of the considered methods for a regularized lid-driven cavity problem

Stabilized reduced basis methods for the approximation of parametrized viscous flows

In Reduced Basis (RB) method, the Galerkin projection on the reduced space does not guarantee the inf-sup approximation stability even if the stable Taylor-Hood Finite Element pair is chosen. Therefore in this PhD thesis we aim to build a stabilized RB method suitable for the approximation of parametrized viscous flows. Starting from the state of the art we study the residual based stabilization techniques for parametrized viscous flows in a RB setting. We are interested in the approximation of the velocity and pressure. extit{Offline-online} computational splitting is implemented and extit{offline-only stabilization}, and extit{offline-online stabilization} are compared (as well as without a stabilization approach). Different test cases are illustrated and several classical stabilization approaches like Brezzi-Pitkaranta, Franca-Hughes, streamline upwind Petrov-Galerkin, Galerkin Least Square are recast into a parametric reduced order setting. The RB method is introduced as a Galerkin projection into reduced spaces, generated by basis functions chosen through a greedy (steady cases) and POD-greedy (unsteady cases) algorithms. This approach is then compared with the supremizer options to guarantee the approximation stability by increasing the corresponding parametric inf-sup condition. We also implement a rectification method to correct the consistency of extit{offline-only stabilization} approach. Several numerical results for both steady and unsteady problems are presented and compared. The goal is two-fold: to guarantee the RB inf-sup stability and to guarantee online computational savings by reducing the dimension of the online reduced basis system

Variational multiscale error estimators for the adaptive mesh refinement of compressible flow simulations

This article investigates an explicit a-posteriori error estimator for the finite element approximation of the compressible Navier–Stokes equations. The proposed methodology employs the Variational Multi-Scale framework, and specifically, the idea is to use the variational subscales to estimate the error. These subscales are defined to be orthogonal to the finite element space, dynamic and non-linear, and both the subscales in the interior of the element and on the element boundaries are considered. Another particularity of the model is that we define some norms that lead to a dimensionally consistent measure of the compressible flow solution error inside each element; a scaled -norm, and the calculation of a physical entropy measure, are both studied in this work. The estimation of the error is used to drive the adaptive mesh refinement of several compressible flow simulations. Numerical results demonstrate good accuracy of the local error estimate and the ability to drive the adaptative mesh refinement to minimize the error through the computational domain.Peer ReviewedPostprint (author's final draft