555 research outputs found
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
An adaptive SUPG method for evolutionary convection-diffusion equations
An adaptive algorithm for the numerical simulation of time-dependent convection-diffusion-reaction equations will be proposed and studied. The algorithm allows the use of the natural extension of any error estimator for the steady-state problem for controlling local refinement and coarsening. The main idea consists in considering the SUPG solution of the evolutionary problem as the SUPG solution of a particular steady-state convection-diffusion problem with data depending on the computed solution. The application of the error estimator is based on a heuristic argument by considering a certain term to be of higher order. This argument is supported in the one-dimensional case by numerical analysis. In the numerical studies, particularly the residual-based error estimator from [18] will be applied, which has proved to be robust in the SUPG norm. The effectivity of this error estimator will be studied and the numerical results (accuracy of the solution, fineness of the meshes) will be compared with results obtained by utilizing the adaptive algorithm proposed in [6]
Primal dual mixed finite element methods for indefinite advection--diffusion equations
We consider primal-dual mixed finite element methods for the
advection--diffusion equation. For the primal variable we use standard
continuous finite element space and for the flux we use the Raviart-Thomas
space. We prove optimal a priori error estimates in the energy- and the
-norms for the primal variable in the low Peclet regime. In the high
Peclet regime we also prove optimal error estimates for the primal variable in
the norm for smooth solutions. Numerically we observe that the method
eliminates the spurious oscillations close to interior layers that pollute the
solution of the standard Galerkin method when the local Peclet number is high.
This method, however, does produce spurious solutions when outflow boundary
layer presents. In the last section we propose two simple strategies to remove
such numerical artefacts caused by the outflow boundary layer and validate them
numerically.Comment: 25 pages, 6 figures, 5 table
A streamline derivative POD-ROM for advection-diffusion-reaction equations
We introduce a new streamline derivative projection-based closure modeling strategy for the numerical stabilization of Proper Orthogonal Decomposition-Reduced Order Models (PODROM). As a first preliminary step, the proposed model is analyzed and tested for advection-dominated advection-diffusion-reaction equations. In this framework, the numerical analysis for the Finite Element (FE) discretization of the proposed new POD-ROM is presented, by mainly deriving the corresponding error estimates. Numerical tests for advection-dominated regime show the efficiency of the proposed method, as well the increased accuracy over the standard POD-ROM that discovers its well-known limitations very soon in the numerical settings considered, i.e. for low diffusion coefficients.Nous introduisons une nouvelle stratégie de modélisation de type streamline derivative basée sur projection pour la stabilisation numérique de modèles d’ordre réduit de type POD (PODROM). Comme première étape préliminaire, le modèle proposé est analysé et testé pour les équations d’advection-diffusion-réaction dominées par l’advection. Dans ce cadre, l’analyse numérique de la discrétisation par éléments finis (FE) du nouveau POD-ROM proposé est présentée, en dérivant principalement les estimations d’erreur correspondantes. Des tests numériques pour le régime dominé par l’advection montrent l’efficacité de la méthode proposée, ainsi que la précision accrue par rapport à la méthode POD-ROM standard qui d´ecouvre très rapidement ses limites bien connues dans le cas des paramètres numériques considérés, c’est-à -dire pour de faibles coefficients de diffusion
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
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