A posteriori error estimations for mixed finite-element approximations to the Navier–Stokes equations

AbstractA posteriori estimates for mixed finite element discretizations of the Navier–Stokes equations are derived. We show that the task of estimating the error in the evolutionary Navier–Stokes equations can be reduced to the estimation of the error in a steady Stokes problem. As a consequence, any available procedure to estimate the error in a Stokes problem can be used to estimate the error in the nonlinear evolutionary problem. A practical procedure to estimate the error based on the so-called postprocessed approximation is also considered. Both the semidiscrete (in space) and the fully discrete cases are analyzed. Some numerical experiments are provided

Pressure-induced locking in mixed methods for time-dependent (Navier-)Stokes equations

We consider inf-sup stable mixed methods for the time-dependent incompressible Stokes and Navier--Stokes equations, extending earlier work on the steady (Navier-)Stokes Problem. A locking phenomenon is identified for classical inf-sup stable methods like the Taylor-Hood or the Crouzeix-Raviart elements by a novel, elegant and simple numerical analysis and corresponding numerical experiments, whenever the momentum balance is dominated by forces of a gradient type. More precisely, a reduction of the $L^2$ convergence order for high order methods, and even a complete stall of the $L^2$ convergence order for lowest-order methods on preasymptotic meshes is predicted by the analysis and practically observed. On the other hand, it is also shown that (structure-preserving) pressure-robust mixed methods do not suffer from this locking phenomenon, even if they are of lowest-order. A connection to well-balanced schemes for (vectorial) hyperbolic conservation laws like the shallow water or the compressible Euler equations is made.Comment: 5 page

Two-Grid Mixed Finite-Element Approximations to the Navier–Stokes Equations Based on a Newton-Type Step

This is post-peer-review, pre-copyedit version of an article published in Journal of Scientific Computing. The final of authenticated version is available online at: https://doi.org/10.1007/s10915-017-0447-2A two-grid scheme to approximate the evolutionary Navier–Stokes equations is introduced and analyzed. A standard mixed finite element approximation is first obtained over a coarse mesh of size H at any positive time T>0 . Then, the approximation is postprocessed by means of solving a steady problem based on one step of a Newton iteration over a finer mesh of size h<H . The method increases the rate of convergence of the standard Galerkin method in one unit in terms of H and equals the rate of convergence of the standard Galerkin method over the fine mesh h. However, the computational cost is essentially the cost of approaching the Navier–Stokes equations with the plain Galerkin method over the coarse mesh of size H since the cost of solving one single steady problem is negligible compared with the cost of computing the Galerkin approximation over the full time interval (0, T]. For the analysis we take into account the loss of regularity at initial time of the solution of the Navier–Stokes equations in the absence of nonlocal compatibility conditions. Some numerical experiments are shownJ. Novo: Research supported by Spanish MINECO under grants MTM2013-42538-P (MINECO, ES) and MTM2016-78995-P (AEI/FEDER, UE

A posteriori error estimation for incompressible viscous fluid with a new boundary condition

This paper describes numerical solutions of incompressible Navier-Stokes equations with a new boundary condition. To solve this problem, we use the discretization by mixed finite element method. We use a vector extrapolation method for computing numerical solutions of the steady-state Navier-Stokes equations. In addition, two types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true error. A numerical experiment on the driven cavity flow is given to demonstrate the effectiveness of the vector extrapolation method. We compare the result with the solution from commercial code like ADINA system as well as with values from other simulations

Preconditioning of Hybridizable Discontinuous Galerkin Discretizations of the Navier-Stokes Equations

The incompressible Navier-Stokes equations are of major interest due to their importance in modelling fluid flow problems. However, solving the Navier-Stokes equations is a difficult task. To address this problem, in this thesis, we consider fast and efficient solvers. We are particularly interested in solving a new class of hybridizable discontinuous Galerkin (HDG) discretizations of the incompressible Navier-Stokes equations, as these discretizations result in exact mass conservation, are locally conservative, and have fewer degrees of freedom than discontinuous Galerkin methods (which is typically used for advection dominated flows). To achieve this goal, we have made various contributions to related problems, as I discuss next. Firstly, we consider the solution of matrices with 2x2 block structure. We are interested in this problem as many discretizations of the Navier-Stokes equations result in block linear systems of equations, especially discretizations based on mixed-finite element methods like HDG. These systems also arise in other areas of computational mathematics, such as constrained optimization problems, or the implicit or steady state treatment of any system of PDEs with multiple dependent variables. Often, these systems are solved iteratively using Krylov methods and some form of block preconditioner. Under the assumption that one diagonal block is inverted exactly, we prove a direct equivalence between convergence of 2x2 block preconditioned Krylov or fixed-point iterations to a given tolerance, with convergence of the underlying preconditioned Schur-complement problem. In particular, results indicate that an effective Schur-complement preconditioner is a necessary and sufficient condition for rapid convergence of 2x2 block-preconditioned GMRES, for arbitrary relative-residual stopping tolerances. A number of corollaries and related results give new insight into block preconditioning, such as the fact that approximate block-LDU or symmetric block-triangular preconditioners offer minimal reduction in iteration over block-triangular preconditioners, despite the additional computational cost. We verify the theoretical results numerically on an HDG discretization of the steady linearized Navier--Stokes equations. The findings also demonstrate that theory based on the assumption of an exact inverse of one diagonal block extends well to the more practical setting of inexact inverses. Secondly, as an initial step towards solving the time-dependent Navier-Stokes equations, we investigate the efficiency, robustness, and scalability of approximate ideal restriction (AIR) algebraic multigrid as a preconditioner in the all-at-once solution of a space-time HDG discretization of the scalar advection-diffusion equation. The motivation for this study is two-fold. First, the HDG discretization of the velocity part of the momentum block of the linearized Navier-Stokes equations is the HDG discretization of the vector advection-diffusion equation. Hence, efficient and fast solution of the advection-diffusion problem is a prerequisite for developing fast solvers for the Navier-Stokes equations. The second reason to study this all-at-once space-time problem is that the time-dependent advection-diffusion equation can be seen as a ``steady'' advection-diffusion problem in (d+1)-dimensions and AIR has been shown to be a robust solver for steady advection-dominated problems. We present numerical examples which demonstrate the effectiveness of AIR as a preconditioner for time-dependent advection-diffusion problems on fixed and time-dependent domains, using both slab-by-slab and all-at-once space-time discretizations, and in the context of uniform and space-time adaptive mesh refinement. A closer look at the geometric coarsening structure that arises in AIR also explains why AIR can provide robust, scalable space-time convergence on advective and hyperbolic problems, while most multilevel parallel-in-time schemes struggle with such problems. As the final topic of this thesis, we extend two state-of-the-art preconditioners for the Navier-Stokes equations, namely, the pressure convection-diffusion and the grad-div/augmented Lagrangian preconditioners to HDG discretizations. Our preconditioners are simple to implement, and our numerical results show that these preconditioners are robust in h and only mildly dependent on the Reynolds numbers

Collision in a cross-shaped domain --- A steady 2D Navier--Stokes example demonstrating the importance of mass conservation in CFD

In the numerical simulation of the incompressible Navier-Stokes equations different numerical instabilities can occur. While instability in the discrete velocity due to dominant convection and instability in the discrete pressure due to a vanishing discrete LBB constant are well-known, instability in the discrete velocity due to a poor mass conservation at high Reynolds numbers sometimes seems to be underestimated. At least, when using conforming Galerkin mixed finite element methods like the Taylor-Hood element, the classical grad-div stabilization for enhancing discrete mass conservation is often neglected in practical computations. Though simple academic flow problems showing the importance of mass conservation are well-known, these examples differ from practically relevant ones, since specially designed force vectors are prescribed. Therefore we present a simple steady Navier-Stokes problem in two space dimensions at Reynolds number 1024, a colliding flow in a cross-shaped domain, where the instability of poor mass conservation is studied in detail and where no force vector is prescribed

Collision in a cross-shaped domain

In the numerical simulation of the incompressible Navier-Stokes equations different numerical instabilities can occur. While instability in the discrete velocity due to dominant convection and instability in the discrete pressure due to a vanishing discrete LBB constant are well-known, instability in the discrete velocity due to a poor mass conservation at high Reynolds numbers sometimes seems to be underestimated. At least, when using conforming Galerkin mixed finite element methods like the Taylor-Hood element, the classical grad-div stabilization for enhancing discrete mass conservation is often neglected in practical computations. Though simple academic flow problems showing the importance of mass conservation are well-known, these examples differ from practically relevant ones, since specially designed force vectors are prescribed. Therefore we present a simple steady Navier-Stokes problem in two space dimensions at Reynolds number 1024, a colliding flow in a cross-shaped domain, where the instability of poor mass conservation is studied in detail and where no force vector is prescribed

On the stability of projection methods for the incompressible Navier-Stokes equations based on high-order discontinuous Galerkin discretizations

The present paper deals with the numerical solution of the incompressible Navier-Stokes equations using high-order discontinuous Galerkin (DG) methods for discretization in space. For DG methods applied to the dual splitting projection method, instabilities have recently been reported that occur for coarse spatial resolutions and small time step sizes. By means of numerical investigation we give evidence that these instabilities are related to the discontinuous Galerkin formulation of the velocity divergence term and the pressure gradient term that couple velocity and pressure. Integration by parts of these terms with a suitable definition of boundary conditions is required in order to obtain a stable and robust method. Since the intermediate velocity field does not fulfill the boundary conditions prescribed for the velocity, a consistent boundary condition is derived from the convective step of the dual splitting scheme to ensure high-order accuracy with respect to the temporal discretization. This new formulation is stable in the limit of small time steps for both equal-order and mixed-order polynomial approximations. Although the dual splitting scheme itself includes inf-sup stabilizing contributions, we demonstrate that spurious pressure oscillations appear for equal-order polynomials and small time steps highlighting the necessity to consider inf-sup stability explicitly.Comment: 31 page