Stability and approximability of the 𝒫1–𝒫0 element for Stokes equations

In this paper we study the stability and approximability of the ??₁–??₀ element (continuous piecewise linear for the velocity and piecewise constant for the pressure on triangles) for Stokes equations. Although this element is unstable for all meshes, it provides optimal approximations for the velocity and the pressure in many cases. We establish a relation between the stabilities of the ??₁–??₀ element (bilinear/constant on quadrilaterals) and the ??₁–??₀ element. We apply many stability results on the ??₁–??₀ element to the analysis of the ??₁–??₀ element. We prove that the element has the optimal order of approximations for the velocity and the pressure on a variety of mesh families. As a byproduct, we also obtain a basis of divergence‐free piecewise linear functions on a mesh family on squares. Numerical tests are provided to support the theory and to show the efficiency of the newly discovered, truly divergence‐free, ??₁ finite element spaces in computation. Copyright © 2006 John Wiley & Sons, Ltd.     

Stability of some low‐order approximations for the Stokes problem

Two‐level low‐order finite element approximations are considered for the inhomogeneous Stokes equations. The elements introduced are attractive because of their simplicity and computational efficiency. In this paper, the stability of a Q₁(h)–Q₁(2h) approximation is analysed for general geometries. Using the macroelement technique, we prove the stability condition for both two‐ and three‐dimensional problems. As a result, optimal rates of convergence are found for the velocity and pressure approximations. Numerical results for three test problems are presented. We observe that for the computed examples, the accuracy of the two‐level bilinear approximation is compared favourably with some standard finite elements. Copyright © 2007 John Wiley & Sons, Ltd.     

High‐order 𝒞1 finite‐element interpolating schemes—Part II: Nonlinear semi‐Lagrangian shallow‐water models

The finite‐element, semi‐implicit, and semi‐Lagrangian methods are used on unstructured meshes to solve the nonlinear shallow‐water system. Several ??¹ approximation schemes are developed for an accurate treatment of the advection terms. The employed finite‐element discretization schemes are the P–P₁ and P₂–P₁ pairs. Triangular finite elements are attractive because of their flexibility for representing irregular boundaries and for local mesh refinement. By tracking the characteristics backward from both the interpolation and quadrature nodes and using ??¹ interpolating schemes, an accurate treatment of the nonlinear terms and, hence, of Rossby waves is obtained. Results of test problems to simulate slowly propagating Rossby modes illustrate the promise of the proposed approach in ocean modelling. Copyright © 2007 John Wiley & Sons, Ltd.     

Stabilized finite element formulation for incompressible flow on distorted meshes

Flow computations frequently require unfavourably meshes, as for example highly stretched elements in regions of boundary layers or distorted elements in deforming arbitrary Lagrangian Eulerian meshes. Thus, the performance of a flow solver on such meshes is of great interest. The behaviour of finite elements with residual‐based stabilization for incompressible Newtonian flow on distorted meshes is considered here. We investigate the influence of the stabilization terms on the results obtained on distorted meshes by a number of numerical studies. The effect of different element length definitions within the elemental stabilization parameter is considered. Further, different variants of residual‐based stabilization are compared indicating that dropping the second derivatives from the stabilization operator, i.e. using a streamline upwind Petrov–Galerkin type of formulation yields better results in a variety of cases. A comparison of the performance of linear and quadratic elements reveals further that the inconsistency of linear elements equipped with residual‐based stabilization introduces significant errors on distorted meshes, while quadratic elements are almost unaffected by moderate mesh distortion. Copyright © 2008 John Wiley & Sons, Ltd.     

On the quadrilateral Q2–P1 element for the Stokes problem

The Q₂ – P₁ approximation is one of the most popular Stokes elements. Two possible choices are given for the definition of the pressure space: one can either use a global pressure approximation (that is on each quadrilateral the finite element space is spanned by 1 and by the global co‐ordinates x and y) or a local approach (consisting in generating the local space by means of the constants and the local curvilinear co‐ordinates on each quadrilateral ξ and η). The former choice is known to provide optimal error estimates on general meshes. This has been shown, as it is standard, by proving a discrete inf–sup condition. In the present paper we check that the latter approach satisfies the inf–sup condition as well. However, recent results on quadrilateral finite elements bring to light a lack in the approximation properties for the space coming out from the local pressure approach. Numerical results actually show that the second choice (local or mapped pressure approximation) is suboptimally convergent. Copyright © 2002 John Wiley & Sons, Ltd.     

A preconditioned conjugate gradient Uzawa-type method for the solution of the stokes problem by mixed Q1–P0 stabilized finite elements

We study the behaviour of a conjugate gradient Uzawa-type method for a stabilized finite element approximation of the Stokes problem. Many variants of the Uzawa algorithm have been described for different finite elements satisfying the well-known Inf-Sup condition of Babu?ka and Brezzi, but it is surprising that developments for unstable ‘low-order’ discretizations with stabilization procedures are still missing. Our paper is presented in this context for the popular (so-called) Q1–P0 element. First we show that a simple stabilization technique for this element permits us to retain the property of a convergence factor bounded independently of the discretization mesh size. The second contribution of this work deals with the construction of a less costly preconditioner taking full advantages of the block diagonal structure of the stabilization matrix. Its efficiency is supported by 2D and. 3D numerical results.     

On pressure separation algorithms (PSepA) for improving the accuracy of incompressible flow simulations

We investigate a special technique called ‘pressure separation algorithm’ (PSepA) (see Applied Mathematics and Computation 2005; 165 :275–290 for an introduction) that is able to significantly improve the accuracy of incompressible flow simulations for problems with large pressure gradients. In our numerical studies with the computational fluid dynamics package FEATFLOW ( www.featflow.de ), we mainly focus on low‐order Stokes elements with nonconforming finite element approximations for the velocity and piecewise constant pressure functions. However, preliminary numerical tests show that this advantageous behavior can also be obtained for higher‐order discretizations, for instance, with Q₂/P₁ finite elements. We analyze the application of this simple, but very efficient, algorithm to several stationary and nonstationary benchmark configurations in 2D and 3D (driven cavity and flow around obstacles), and we also demonstrate its effect to spurious velocities in multiphase flow simulations (‘static bubble’ configuration) if combined with edge‐oriented, resp., interior penalty finite element method stabilization techniques. Copyright © 2008 John Wiley & Sons, Ltd.     

Singular finite elements for the sudden-expansion and the die-swell problems

The singular finite element method is used to solve the sudden-expansion and the die-swell problems in order to improve the accuracy of the solution in the vicinity of the singularity and to speed up the convergence. The method requires minor modifications to standard finite element schemes, and even coarse meshes give more accurate results than refined ordinary finite element meshes. Improved normal stress results for the sudden-expansion problem have been obtained for various Reynolds numbers up to 100 using the singular elements constructed for the creeping flow problem. In addition, the normal stresses at the walls appear to be insensitive to the singularity powers used in the construction of the singular basis functions. The die-swell problem is solved using the singular elements constructed for the stick–slip problem. The singular elements accelerate the convergence of the free surface dramatically.     

Finite element solution of the Orr–Sommerfeld equation using high precision Hermite elements: plane Poiseuille flow

This paper presents a comprehensive review of the numerical techniques used during the past half century and their accuracy in hydrodynamic stability analysis of plane parallel flows. The paper also describes a finite element solution of the Orr–Sommerfeld equation using high precision Hermite elements. A stability analysis technique is performed by imposing an infinitesimal perturbation to the laminar base flow to determine the thresholds of neutral instabilities or the growth rate of the perturbation for any Reynolds and wave numbers. Validation of the present numerical technique is performed for plane Poiseuille flow. The numerical results, obtained with uniform and nonuniform meshes, show excellent agreement with the most accurate results available in the literature. Copyright © 2004 John Wiley & Sons, Ltd.     

Stabilized finite‐element computation of compressible flow with linear and quadratic interpolation functions

The unsteady compressible flow equations are solved using a stabilized finite‐element formulation with C⁰ elements. In 2D, the performance of three‐noded linear and six‐noded quadratic triangular elements is compared. In 3D, the relative performance is evaluated for 6‐noded linear and 18‐noded quadratic wedge elements. Results are compared for the solutions to Euler, laminar, and turbulent flows at different Mach numbers for several flow problems. The finite‐element meshes considered for comparison have same location of nodes for the linear and quadratic interpolations. For the turbulent flow, the Spalart–Allmaras model is used for closure. It is found that the quadratic elements yield better performance than the linear elements. This is attributed to accurate representation of the stabilization terms that involve second‐order derivatives in the formulation. When these terms are dropped from the formulation with quadratic interpolation, the numerical results are similar to those obtained with linear interpolation. The absence of these terms result in added numerical diffusion that seems to give the effect of a relatively reduced Reynolds number. For the same location of nodes, the computations with the linear triangular and wedge elements are approximately 20% and 100% faster than those with quadratic triangular and wedge elements, respectively. However, if the same quadrature rule for numerical integration is used for both interpolations, the computations with quadratic elements are approximately 20% and 45% faster in 2D and 3D, respectively. Copyright © 2014 John Wiley & Sons, Ltd.     

New stabilized finite element method for time‐dependent incompressible flow problems

A new stabilized finite element method is considered for the time‐dependent Stokes problem, based on the lowest‐order P₁?P₀ and Q₁?P₀ elements that do not satisfy the discrete inf–sup condition. The new stabilized method is characterized by the features that it does not require approximation of the pressure derivatives, specification of mesh‐dependent parameters and edge‐based data structures, always leads to symmetric linear systems and hence can be applied to existing codes with a little additional effort. The stability of the method is derived under some regularity assumptions. Error estimates for the approximate velocity and pressure are obtained by applying the technique of the Galerkin finite element method. Some numerical results are also given, which show that the new stabilized method is highly efficient for the time‐dependent Stokes problem. Copyright © 2009 John Wiley & Sons, Ltd.     

On stabilized space‐time FEM for anisotropic meshes: Incompressible Navier–Stokes equations and applications to blood flow in medical devices

In complex applications, such as the analysis of hydraulic performance of blood pumps (ventricular assist devices), the Navier–Stokes equations have to be discretized on very anisotropic meshes. If stabilized finite element formulations are applied, standard definitions of the stabilization parameter are usually not appropriate to handle elements with a high aspect ratio. If, in addition, rotating objects, moving meshes, or turbulence has to be considered in the simulation, further modifications of the stabilization procedure have to be applied. In this paper, we present stabilized space‐time finite element formulations of the incompressible Navier–Stokes equations that show very good convergence properties on complex anisotropic meshes and lead to reasonable numerical accuracy in complex flows when compared with experimental data. Copyright © 2017 John Wiley & Sons, Ltd.     

A control volume finite element method for three-dimensional three-phase flows

A novel control volume finite element method with adaptive anisotropic unstructured meshes is presented for three-dimensional three-phase flows with interfacial tension. The numerical framework consists of a mixed control volume and finite element formulation with a new P₁DG-P₂ elements (linear discontinuous velocity between elements and quadratic continuous pressure between elements). A “volume of fluid” type method is used for the interface capturing, which is based on compressive control volume advection and second-order finite element methods. A force-balanced continuum surface force model is employed for the interfacial tension on unstructured meshes. The interfacial tension coefficient decomposition method is also used to deal with interfacial tension pairings between different phases. Numerical examples of benchmark tests and the dynamics of three-dimensional three-phase rising bubble, and droplet impact are presented. The results are compared with the analytical solutions and previously published experimental data, demonstrating the capability of the present method.     

A technique for analysing finite element methods for viscous incompressible flow

We give a self-contained presentation of our macroelement technique for verifying the stability of finite element discretizations of the Navier–Stokes equations in the velocity–pressure formulation.     

High‐order methods for the numerical solution of the BiGlobal linear stability eigenvalue problem in complex geometries

A high‐order computational tool based on spectral and spectral/hp elements (J. Fluid. Mech. 2009; to appear) discretizations is employed for the analysis of BiGlobal fluid instability problems. Unlike other implementations of this type, which use a time‐stepping‐based formulation (J. Comput. Phys. 1994; 110 (1):82–102; J. Fluid Mech. 1996; 322 :215–241), a formulation is considered here in which the discretized matrix is constructed and stored prior to applying an iterative shift‐and‐invert Arnoldi algorithm for the solution of the generalized eigenvalue problem. In contrast to the time‐stepping‐based formulations, the matrix‐based approach permits searching anywhere in the eigenspace using shifting. Hybrid and fully unstructured meshes are used in conjunction with the spatial discretization. This permits analysis of flow instability on arbitrarily complex 2‐D geometries, homogeneous in the third spatial direction and allows both mesh (h)‐refinement as well as polynomial (p)‐refinement. A series of validation cases has been defined, using well‐known stability results in confined geometries. In addition new results are presented for ducts of curvilinear cross‐sections with rounded corners. Copyright © 2010 John Wiley & Sons, Ltd.     

The influence of the stabilization parameter on the convergence factor of iterative methods for the solution of the discretized Stokes problem

This paper discusses the influence of the stabilization parameter on the convergence factor of various iterative methods for the solution of the Stokes problem discretized by the so-called locally stabilized Q1-P0 finite element. Our objective is to point out optimal parameters which ensure rapid convergence. The first part of the paper is concerned with the dual formulation of the problem. It gives the theoretical precision and practical developments of our stabilized context Uzawa-type algorithm. We assert that the convergence factor of such a method is majored independently of the mesh size by a function of the stabilization parameter. Moreover, we point out that there exists an optimal value of this parameter that minimizes this upper bound. This gives a theoretical justification of pre-existing numerical results. We show that the optimal parameter can be determined a priori. This is a key point when the method has to be implemented. Finally, we base an interpretation of the iterated penalty method numerical behaviour on some theoretical results about the minimum eigenvalue of the stabilized dual operator. This algorithm involves a penalty parameter and a stabilization parameter and we discuss a strategy for choosing optimal parameters. The mixed formulation of the problem is dealt with in the second part of the paper, which proposes several preconditioned conjugate-gradient-type methods. The indefinite character of the problem makes it intrinsically hard. However, if one chooses a suitable preconditioner, this difficulty is overcome, since the preconditioned operator becomes positive definite. We study the eigenvalue spectrum of the preconditioned operator and thereby the convergence factor of the algorithm. In contrast with the two previous formulations, we show that this convergence factor is majored independently of the stabilization parameter. More precisely, we point out convergence factors comparable with those obtained for Poisson-type problems. Finally, we present a variant of the latter method which uses our so-called macroblock-type preconditioner. A comparison with the simple case of diagonal preconditioning is addressed and the improved performance of the macroblock-type preconditioner is evidenced. Various 2D numerical experiments are given to corroborate the theories presented herein.     

A Taylor–Galerkin-based algorithm for viscous incompressible flow

In this paper the development and behaviour of a new finite element algorithm for viscous incompressible flow is presented. The stability and background theory are discussed and the numerical performance is considered for some benchmark problems. The Taylor–Galerkin approach naturally leads to a time-stepping algorithm which is shown to perform well for a wide range of Reynolds numbers (1 ? Re ? 400).

1 A conventional definition for Re is assumed.

Various modifications to the algorithm are investigated, particularly with respect to their effects on stability and accuracy.     

A sensitivity analysis on the parameter of the GLS method for a second‐gradient theory of incompressible flow

Using a non‐conforming C⁰‐interior penalty method and the Galerkin least‐square approach, we develop a continuous–discontinuous Galerkin finite element method for discretizing fourth‐order incompressible flow problems. The formulation is weakly coercive for spaces that fail to satisfy the inf‐sup condition and consider discontinuous basis functions for the pressure field. We consider the results of a stability analysis through a lemma which indicates that there exists an optimal or quasi‐optimal least‐square stability parameter that depends on the polynomial degree used to interpolate the velocity and pressure fields, and on the geometry of the finite element in the mesh. We provide several numerical experiments illustrating such dependence, as well as the robustness of the method to deal with arbitrary basis functions for velocity and pressure, and the ability to stabilize large pressure gradients. We believe the results provided in this paper contribute for establishing a paradigm for future studies of the parameter of the Galerkin least square method for second‐gradient theory of incompressible flow problems. Copyright © 2015 John Wiley & Sons, Ltd.     

An edge-based pressure stabilization technique for finite elements on arbitrarily anisotropic meshes

In this paper, we analyze a stabilized equal-order finite element approximation for the Stokes equations on anisotropic meshes. In particular, we allow arbitrary anisotropies in a subdomain, for example, along the boundary of the domain, with the only condition that a maximum angle is fulfilled in each element. This discretization is motivated by applications on moving domains as arising, for example, in fluid-structure interaction or multiphase-flow problems. To deal with the anisotropies, we define a modification of the original continuous interior penalty stabilization approach. We show analytically the discrete stability of the method and convergence of order in the energy norm and in the L²-norm of the velocities. We present numerical examples for a linear Stokes problem and for a nonlinear fluid-structure interaction problem, which substantiate the analytical results and show the capabilities of the approach.     

Numerical investigations of stabilized finite element computations for acoustics

Least-squares stabilization stands out among the numerous approaches that have been proposed for relaxing resolution requirements of Galerkin computations for acoustics, by combining substantial improvement in performance with extremely simple implementation. The Galerkin/least-squares and Galerkin-gradient/least-squares methods are quite similar for structured meshes of linear finite elements. A series of numerical tests compares the two methods for several configurations with different kinds of boundary conditions employing structured and unstructured meshes. Various definitions of the resolution-dependent stability parameters are considered, along with different definitions of the mesh size upon which they depend.