Local Projection Stabilization for the Oseen Problem and its Interpretation as a Variational Multiscale Method (original) (raw)
Related papers
Stabilized finite element methods for the generalized Oseen problem
Computer Methods in Applied Mechanics and Engineering, 2007
The numerical solution of the non-stationary, incompressible Navier-Stokes model can be split into linearized auxiliary problems of Oseen type. We present in a unique way different stabilization techniques of finite element schemes on isotropic meshes. First we describe the state-of-the-art for the classical residual-based SUPG/PSPG method. Then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method. These methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation. Finally, we give a critical comparison of these methods.
A variational multiscale stabilized formulation for the incompressible Navier–Stokes equations
Computational Mechanics, 2009
This paper presents a variational multiscale residual-based stabilized finite element method for the incompressible Navier-Stokes equations. Structure of the stabilization terms is derived based on the two level scale separation furnished by the variational multiscale framework. A significant feature of the new method is that the fine scales are solved in a direct nonlinear fashion, and a definition of the stabilization tensor τ is derived via the solution of the fine-scale problem. A computationally economic procedure is proposed to evaluate the advection part of the stabilization tensor. The new method circumvents the Babuska-Brezzi (inf-sup) condition and yields a stable formulation for high Reynolds number flows. A family of equal-order pressurevelocity elements comprising 4-and 10-node tetrahedral elements and 8-and 27-node hexahedral elements is developed. Convergence rates are reported and accuracy properties of the method are presented via the lid-driven cavity flow problem. Keywords Multiscale finite element methods • Navier-Stokes equations • Convergence rates • Equal order interpolation functions • Tetrahedral elements • Hexahedral elements
A unified convergence analysis for local projection stabilisations applied to the Oseen problem
ESAIM: Mathematical Modelling and Numerical Analysis, 2007
The discretisation of the Oseen problem by finite element methods may suffer in general from two shortcomings. First, the discrete inf-sup (Babuška-Brezzi) condition can be violated. Second, spurious oscillations occur due to the dominating convection. One way to overcome both difficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we show that the fulfilment of a local inf-sup condition between approximation and projection spaces allows to construct an interpolation with additional orthogonality properties. Based on this special interpolation, optimal a-priori error estimates are shown with error constants independent of the Reynolds number. Applying the general theory, we extend the results of Braack and Burman for the standard two-level version of the local projection stabilisation to discretisations of arbitrary order on simplices, quadrilaterals, and hexahedra. Moreover, our general theory allows to derive a novel class of local projection stabilisation by enrichment of the approximation spaces. This class of stabilised schemes uses approximation and projection spaces defined on the same mesh and leads to much more compact stencils than in the two-level approach. Finally, on simplices, the spectral equivalence of the stabilising terms of the local projection method and the subgrid modelling introduced by Guermond is shown. This clarifies the relation of the local projection stabilisation to the variational multiscale approach.
Computer Methods in Applied Mechanics and Engineering, 2016
The variational multiscale method thought as an implicit large eddy simulation model for turbulent flows has been shown to be an alternative to the widely used physical-based models. This method is traditionally combined with equal-order velocity-pressure pairs, since it provides pressure stabilization. In this work, we consider a different approach, based on inf-sup stable elements and convection-only stabilization. In order to do so, we consider a symmetric projection stabilization of the convective term using a orthogonal subscale decomposition. The accuracy and efficiency of this method compared with residual-based algebraic subgrid scales and orthogonal subscales methods for equal-order interpolation is assessed in this paper. Moreover, when inf-sup stable elements are used, the grad-div stabilization term has been shown to be essential to guarantee accurate solutions. Hence, a study of the influence of such term in the large eddy simulation of turbulent incompressible flows is also performed. Furthermore, a recursive block preconditioning strategy has been considered for the resolution of the problem with an implicit treatment of the projection terms. Two different benchmark tests have been solved: the Taylor-Green Vortex flow with Re = 1600, and the Turbulent Channel Flow at Re τ = 395.
Large scale finite element solvers for the large eddy simulation of incompressible turbulent flows
2016
In this thesis we have developed a path towards large scale Finite Element simulations of turbulent incompressible flows. We have assessed the performance of residual-based variational multiscale (VMS) methods for the large eddy simulation (LES) of turbulent incompressible flows. We consider VMS models obtained by different subgrid scale approximations which include either static or dynamic subscales, linear or nonlinear multiscale splitting, and different choices of the subscale space. We show that VMS thought as an implicit LES model can be an alternative to the widely used physical-based models. This method is traditionally combined with equal-order velocity-pressure pairs, since it provides pressure stabilization. In this work, we also consider a different approach, based on inf-sup stable elements and convection-only stabilization. In order to do so, we define a symmetric projection stabilization of the convective term using an orthogonal subscale decomposition. The accuracy an...
Beyond pressure stabilization: A low-order local projection method for the Oseen equation
International Journal for Numerical Methods in Engineering, 2011
This work proposes a new local projection stabilized finite element method (LPS) for the Oseen problem. The method adds to the Galerkin formulation new fluctuation terms which are symmetric and easily computable at the element level. Proposed for the pair P 1 /P l , l = 0, 1, when the pressure is continuously or discontinuously approximated, wellposedeness and error optimality are proved. In addition, we introduce a cheap strategy to recover an element-wise mass conservative velocity field in the discontinuous pressure case, a property usually neglected in the stabilized finite element context. Numerics validate the theoretical results and show that the present method improves accuracy to represent boundary layers when compared with alternative approaches.
arXiv: Numerical Analysis, 2018
In this paper, we consider up-to-date and classical Finite Element (FE) stabilized methods for time-dependent incompressible flows. All studied methods belong to the Variational MultiScale (VMS) framework. So, different realizations of stabilized FE-VMS methods are compared in high Reynolds numbers vortex dynamics simulations. In particular, a fully Residual-Based (RB)-VMS method is compared with the classical Streamline-Upwind Petrov--Galerkin (SUPG) method together with grad-div stabilization, a standard one-level Local Projection Stabilization (LPS) method, and a recently proposed LPS method by interpolation. These procedures do not make use of the statistical theory of equilibrium turbulence, and no ad-hoc eddy viscosity modeling is required for all methods. Applications to the simulations of high Reynolds numbers flows with vortical structures on relatively coarse grids are showcased, by focusing on two-dimensional plane mixing-layer flows. Both Inf-Sup Stable (ISS) and Equal O...
Mathematical Models and Methods in Applied Sciences, 2012
We consider a projection-based variational multiscale method for large-eddy simulation of the Navier–Stokes/Fourier model of incompressible, non-isothermal flows. For the semidiscrete problem, an a priori error estimate is given for rather general nonlinear, piecewise constant coefficients of the subgrid models for the unresolved scales of velocity, pressure, and temperature. Then we address aspects of the discretization in time. Finally, the design of the subgrid scale models is specified for the case of free convection problems and studied for the standard benchmark problem of free convection in a closed cavity.
Local projection stabilization for the Oseen problem
IMA Journal of Numerical Analysis, 2015
We consider conforming finite element (FE) approximations of the timedependent Oseen problem with inf-sup stable approximation of velocity and pressure. It serves as a preliminary study of the incompressble Navier-Stokes problem. In case of high Reynolds numbers, the local projection stabilization (LPS) method is considered. In particular, the idea of streamline upwinding is combined with stabilization of the divergence-free constraint. For the arising semidiscrete problem, a stability and convergence analysis is given. Our approach improves some results of a recent paper by Matthies/Tobiska [13]. Finally, we apply the approach to the timedependent incompressible Navier-Stokes problem, test the accuracy of the method and conduct numerical experiments with simple boundary layers and separation.