Pairs Of Finite Element Spaces Research Articles

On the convergence order of the finite element error in the kinetic energy for high Reynolds number incompressible flows

The kinetic energy of a flow is proportional to the square of the L2(Ω) norm of the velocity. Given a sufficient regular velocity field and a velocity finite element space with polynomials of degree r, then the best approximation error in L2(Ω) is of order r+1. In this survey, the available finite element error analysis for the velocity error in L∞(0,T;L2(Ω)) is reviewed, where T is a final time. Since in practice the case of small viscosity coefficients or dominant convection is of particular interest, which may result in turbulent flows, robust error estimates are considered, i.e., estimates where the constant in the error bound does not depend on inverse powers of the viscosity coefficient. Methods for which robust estimates can be derived enable stable flow simulations for small viscosity coefficients on comparatively coarse grids, which is often the situation encountered in practice. To introduce stabilization techniques for the convection-dominated regime and tools used in the error analysis, evolutionary linear convection–diffusion equations are studied at the beginning. The main part of this survey considers robust finite element methods for the incompressible Navier–Stokes equations of order r−1, r, and r+1/2 for the velocity error in L∞(0,T;L2(Ω)). All these methods are discussed in detail. In particular, a sketch of the proof for the error bound is given that explains the estimate of important terms which determine finally the order of convergence. Among them, there are methods for inf–sup stable pairs of finite element spaces as well as for pressure-stabilized discretizations. Numerical studies support the analytic results for several of these methods. In addition, methods are surveyed that behave in a robust way but for which only a non-robust error analysis is available. The conclusion of this survey is that the problem of whether or not there is a robust method with optimal convergence order for the kinetic energy is still open.

NewH(div)-conforming multiscale hybrid-mixed methods for the elasticity problem on polygonal meshes

This work proposes a family of multiscale hybrid-mixed methods for the two-dimensional linear elasticity problem on general polygonal meshes. The new methods approximate displacement, stress, and rotation using two-scale discretizations. The first scale level setting consists of approximating the traction variable (Lagrange multiplier) in discontinuous polynomial spaces, and of computing elementwise rigid body modes. In the second level, the methods are made effective by solving completely independent local boundary Neumann elasticity problems written in a mixed form with weak symmetry enforcedviathe rotation multiplier. Since the finite-dimensional space for the traction variable constraints the local stress approximations, the discrete stress field lies in theH(div) space globally and stays in local equilibrium with external forces. We propose different choices to approximate local problems based on pairs of finite element spaces defined on affine second-level meshes. Those choices generate the family of multiscale finite element methods for which stability and convergence are proved in a unified framework. Notably, we prove that the methods are optimal and high-order convergent in the natural norms. Also, it emerges that the approximate displacement and stress divergence are super-convergent in theL2-norm. Numerical verifications assess theoretical results and highlight the high precision of the new methods on coarse meshes for multilayered heterogeneous material problems.

Higher-order discontinuous Galerkin time discretizations for the evolutionary Navier–Stokes equations

Abstract Discontinuous Galerkin methods of higher order are applied as temporal discretizations for the transient Navier–Stokes equations. The spatial discretization based on inf–sup stable pairs of finite element spaces is stabilized using a one-level local projection stabilization method. Optimal error bounds for the velocity with constants independent of the viscosity parameter are obtained for both the semidiscrete case and the fully discrete case. Numerical results support the theoretical predictions.

An assessment of two classes of variational multiscale methods for the simulation of incompressible turbulent flows

A numerical assessment of two classes of variational multiscale (VMS) methods for the simulation of incompressible flows is presented. Two types of residual-based VMS methods and two types of projection-based VMS methods are included in this assessment. The numerical simulations are performed at turbulent channel flow problems with various friction Reynolds numbers. It turns out the residual-based VMS methods, in particular when used with a pair of inf–sup stable finite elements, give usually the most accurate results for second order statistics. For this pair of finite element spaces, a flexible GMRES method with a Least Squares Commutator (LSC) preconditioner proved to be an efficient solver.

Quasi-optimal and pressure-robust discretizations of the Stokes equations by new augmented Lagrangian formulations

Abstract We approximate the solution of the stationary Stokes equations with various conforming and nonconforming inf-sup stable pairs of finite element spaces on simplicial meshes. Based on each pair, we design a discretization that is quasi-optimal and pressure-robust, in the sense that the velocity $H^1$-error is proportional to the best velocity $H^1$-error. This shows that such a property can be achieved without using conforming and divergence-free pairs. We also bound the pressure $L^2$-error, only in terms of the best velocity $H^1$-error and the best pressure $L^2$-error. Our construction can be summarized as follows. First, a linear operator acts on discrete velocity test functions, before the application of the load functional, and maps the discrete kernel into the analytical one. Second, in order to enforce consistency, we employ a new augmented Lagrangian formulation, inspired by discontinuous Galerkin methods.

WITHDRAWN: Note on the usage of grad-div stabilization for the penalty-projection algorithm in magnetohydrodynamics

Abstract An algorithm for resolving magnetohydrodynamic (MHD) flows has been presented recently, that allows for a stable decoupling of the system and uses the penalty-projection method for extra efficiency. The algorithm relies on the choice of Scott–Vogelius finite elements to complement the grad-div stabilization. We propose a small modification of the algorithm, which allows for its usage even with the less sophisticated (and more computationally attractive) Taylor–Hood pair of finite element spaces. We demonstrate numerically, that the new modification of the method is first order accurate in time (as expected by the theory), while the existing method would fail on the Taylor–Hood finite elements (the blow-up of the solution is demonstrated).

Note on the usage of grad-div stabilization for the penalty-projection algorithm in magnetohydrodynamics

An algorithm for resolving magnetohydrodynamic (MHD) flows has been presented recently, that allows for a stable decoupling of the system and uses the penalty-projection method for extra efficiency. The algorithm relies on the choice of Scott–Vogelius finite elements to complement the grad-div stabilization. We propose a small modification of the algorithm, which allows for its usage even with the less sophisticated (and more computationally attractive) Taylor–Hood pair of finite element spaces. We demonstrate numerically, that the new modification of the method is first order accurate in time (as expected by the theory), while the existing method would fail on the Taylor–Hood finite elements (the blow-up of the solution is demonstrated).

Whether or not rehabilitation helps to reduce the risk of developmental delay in children with epilepsy debut under first year of life?

Efficient incompressible flow simulations, using inf–sup stable pairs of finite element spaces, require the application of efficient solvers for the arising linear saddle point problems. This paper presents an assessment of different solvers: the sparse direct solver UMFPACK, the flexible GMRES (FGMRES) method with different coupled multigrid preconditioners, and FGMRES with Least Squares Commutator (LSC) preconditioners. The assessment is performed for steady-state and time-dependent flows around cylinders in 2d and 3d. Several pairs of inf–sup stable finite element spaces with second order velocity and first order pressure are used. It turns out that for the steady-state problems often FGMRES with an appropriate multigrid preconditioner was the most efficient method on finer grids. For the time-dependent problems, FGMRES with LSC preconditioners that use an inexact iterative solution of the velocity subproblem worked best for smaller time steps.

CBS Constants & Their Role in Error Estimation for Stochastic Galerkin Finite Element Methods

Stochastic Galerkin finite element methods (SGFEMs) are commonly used to approximate solutions to PDEs with random inputs. However, the study of a posteriori error estimation strategies to drive adaptive enrichment of the associated tensor product spaces is still under development. In this work, we revisit an a posteriori error estimator introduced in Bespalov and Silvester (SIAM J Sci Comput 38(4):A2118–A2140, 2016) for SGFEM approximations of the parametric reformulation of the stochastic diffusion problem. A key issue is that the bound relating the true error to the estimated error involves a CBS (Cauchy–Buniakowskii–Schwarz) constant. If the approximation spaces associated with the parameter domain are orthogonal in a weighted L^2 sense, then this CBS constant only depends on a pair of finite element spaces H_{1}, H_{2} associated with the spatial domain and their compatibility with respect to an inner product associated with a parameter-free problem. For fixed choices of H_{1}, we investigate non-standard choices of H_{2} and the associated CBS constants, with the aim of designing efficient error estimators with effectivity indices close to one. When H_1 and H_2 satisfy certain conditions, we also prove new theoretical estimates for the CBS constant using linear algebra arguments.

An assessment of some solvers for saddle point problems emerging from the incompressible Navier–Stokes equations

Efficient incompressible flow simulations, using inf–sup stable pairs of finite element spaces, require the application of efficient solvers for the arising linear saddle point problems. This paper presents an assessment of different solvers: the sparse direct solver UMFPACK, the flexible GMRES (FGMRES) method with different coupled multigrid preconditioners, and FGMRES with Least Squares Commutator (LSC) preconditioners. The assessment is performed for steady-state and time-dependent flows around cylinders in 2d and 3d. Several pairs of inf–sup stable finite element spaces with second order velocity and first order pressure are used. It turns out that for the steady-state problems often FGMRES with an appropriate multigrid preconditioner was the most efficient method on finer grids. For the time-dependent problems, FGMRES with LSC preconditioners that use an inexact iterative solution of the velocity subproblem worked best for smaller time steps.

Isogeometric Analysis of the Navier–Stokes–Cahn–Hilliard equations with application to incompressible two-phase flows

In this work, we provide a unified and comparative description of the most prominent phase field based two-phase flow models and present our numerical results of the application of Galerkin-based Isogeometric Analysis (IGA) to incompressible Navier–Stokes–Cahn–Hilliard (NSCH) equations in velocity–pressure–phase field-chemical potential formulation. For the approximation of the velocity and pressure fields, LBB compatible non-uniform rational B-spline spaces are used which can be regarded as smooth generalizations of Taylor–Hood pairs of finite element spaces. The one-step θ-scheme is used for the discretization in time. The static and rising bubble, in addition to the nonlinear Rayleigh–Taylor instability flow problems, are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme.

A hybrid two-step finite element method for flux approximation:a prioriestimates

We present a new two–step method based on the hybridization of mesh sizes in the traditional mixed finite element method. On a coarse mesh, the primary variable is approximated by a standard Galerkin method, whose computational cost is very low. Then, on a fine mesh, an H (div) projection of the dual variable is sought as an accurate approximation for the flux variable. Our method does not rely on the framework of traditional mixed formulations, the choice of pair of finite element spaces is, therefore, free from the requirement of inf-sup stability condition. More precisely, our method is formulated in a fully decoupled manner, still achieving an optimal error convergence order. This leads to a computational strategy much easier and wider to implement than the mixed finite element method. Additionally, the independently posed solution strategy allows to use different meshes as well as different discretization schemes in the calculation of the primary and flux variables. We show that the finer mesh size h can be taken as the square of the coarse mesh size H , or a higher order power with a proper choice of parameter δ . This means that the computational cost for the coarse-grid solution is negligible compared to that for the fine-grid solution. In fact, numerical experiments show an advantage of using our strategy compared to the mixed finite element method. Some guidelines to choose an optimal parameter δ are also given. In addition, our approach is shown to provide an asymptotically exact a posteriori error estimator for the primary variable p in H 1 norm.

Numerical analysis of Darcy problem on surfaces

Surface problems play a key role in several theoretical and applied fields. In this work the main focus is the presentation of a detailed analysis of the approximation of the classical porous media flow problem: the Darcy equation, where the domain is a regular surface. The formulation considers the mixed form and the numerical approximation adopts a classical pair of finite element spaces: piecewise constant for the scalar fields and Raviart–Thomas for vector fields, both written on the tangential space of the surface. The main result is the proof of the order of convergence where the discretization error, due to the finite element approximation, is coupled with a geometrical error. The latter takes into account the approximation of the real surface with a discretized one. Several examples are presented to show the correctness of the analysis, including surfaces with boundary.

Isogeometric Analysis of the Navier–Stokes equations with Taylor–Hood B-spline elements

This paper presents our numerical results of the application of Isogeometric Analysis (IGA) to the velocity–pressure formulation of the steady state as well as to the unsteady incompressible Navier–Stokes equations. For the approximation of the velocity and pressure fields, LBB compatible B-spline spaces are used which can be regarded as smooth generalizations of Taylor–Hood pairs of finite element spaces. The single-step θ-scheme is used for the discretization in time. The lid-driven cavity flow, in addition to its regularized version and flow around cylinder, are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme.

Superconvergence Analysis for the Stable Conforming Rectangular Mixed Finite Elements for the Linear Elasticity Problem

In this paper, we consider the linear elasticity problem based on the Hellinger-Reissner variational principle. An O(h 2 ) order superclose property for the stress and displacement and a global superconvergence result of the displacement are established by employing a Clement interpolation, an integral identity and appropriate postprocessing techniques. In this paper, we consider the mixed finite element (for short MFE) approximation of a stress-displacement system derived from the Hellinger-Reissner variational principle for the linear elasticity problem. As is known to all, the MFE methods require that the pair of finite element spaces satisfying the B-B condition. Although there are a number of well-known stable MFEs for the analogous problems involving vector fields and scalar fields (1), the combination of the symmetry and continuity conditions of the stress field is a substantial additional difficulty. On the other hand, a lot of efforts, dating back four decades, have been devoted to develop stable MFEs for the linear elasticity problem, but no stable MFE scheme with polynomial shape functions are yielded. Not until the year 2002, were there some development in this direction. In (2), a sufficient condition was given and then a family of stable MFEs were constructed with respect to arbitrary triangular meshes, with 24 stress and 6 displacement degrees of freedom for the lowest order element, and an optimal order error estimate was obtained. An analogous family of conforming MFEs based on rectangular meshes were proposed in (3), involving 45 stress and 12 displacement degrees of freedom for the lowest order element. Two nonconforming triangular elements were presented in (4) with 12 degrees of freedom for the stress and 3 degrees of freedom for the displacement. Although many stable elements have been constructed for this problem, they involve too much degrees of freedom. Recently, some more simple elements have been developed. In (5), a group of nonconforming rectangular elements were introduced, with the convergence order of

Local projection type stabilization applied to inf-sup stable discretizations of the Oseen problem

The local projection method is applied to inf-sup stable discretisations of the Oseen problem. Error bounds of order r are proven for known inf-sup stable pairs of finite element spaces which approximate velocity and pressure by elements of order r and r− 1, respectively. In case of a positive reaction coefficient, the error constants are robust with respect to the viscosity but depend on the positive lower bound of the reaction coefficient. Using enriched velocity spaces, error estimates of order r are established which are also robust when both the viscosity and the reaction coefficient tend to zero. Moreover, for certain velocity and pressure approximations by elements of order r, the discrete inf-sup condition holds and a robust error estimate of improved order r+ 1/2 is shown. Numerical results confirm the theoretical convergence results.

On the parameter choice in grad-div stabilization for the Stokes equations

Standard error analysis for grad-div stabilization of inf-sup stable conforming pairs of finite element spaces predicts that the stabilization parameter should be optimally chosen to be 𝒪 ( 1 ) $\mathcal O(1)$ . This paper revisits this choice for the Stokes equations on the basis of minimizing the H 1 ( Ω ) $H^{1}(\Omega )$ error of the velocity and the L 2 ( Ω ) $L^{2}(\Omega )$ error of the pressure. It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle than known so far in the literature. It depends on the used norm, the solution, the family of finite element spaces, and the type of mesh. In particular, the approximation property of the pointwise divergence-free subspace plays a key role. With such an optimal approximation property and with an appropriate choice of the stabilization parameter, estimates for the H 1 ( Ω ) $H^{1}(\Omega )$ error of the velocity are obtained that do not directly depend on the viscosity and the pressure. The minimization of the L 2 ( Ω ) $L^{2}(\Omega )$ error of the pressure requires in many cases smaller stabilization parameters than the minimization of the H 1 ( Ω ) $H^{1}(\Omega )$ velocity error. Altogether, depending on the situation, the optimal stabilization parameter could range from being very small to very large. The analytic results are supported by numerical examples. Applying the analysis to the MINI element leads to proposals for the stabilization parameter which seem to be new.

IsoGeometric Analysis: Stable elements for the 2D Stokes equation

AbstractIn this paper, we discuss the application of IsoGeometric Analysis to incompressible viscous flow problems. We consider, as a prototype problem, the Stokes system and we propose various choices of compatible spline spaces for the approximations to the velocity and the pressure fields. The proposed choices can be viewed as extensions of the Taylor–Hood, Nédélec and Raviart–Thomas pairs of finite element spaces, respectively. We study the stability and convergence properties of each method and discuss the conservation properties of the discrete velocity field in each case. Copyright © 2010 John Wiley & Sons, Ltd.

Stabilized finite element schemes for generalized Oseen systems

AbstractIn recent works [1, 2], advanced approximation schemes for the numerical calculation of compressible viscous flow were developed, analyzed theoretically and applied successfully to benchmark problems. These methods are based on splitting the Poisson–Stokes system (1.1) describing the motion of a viscous compressible fluid into a generalized Oseen problem for the velocity and a hyperbolic transport equation for the density. Highly refined finite element techniques were proposed for the numerical solution of these separated subproblems of simpler structure; cf. [2]. In this paper, error estimates for a SUPG/(PSPG) and grad‐div stabilized finite element approximation of the generalized Oseen problems that arise in the course of the splitting procedure are presented. LBB‐stable pairs of finite element spaces are used for velocity and pressure. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Overlapping Schwarz preconditioner for the mixed formulation of plane elasticity

Recently a stable pair of finite element spaces for the mixed formulation of the plane elasticity system has been developed by Arnold and Winther. Here we construct a two-level overlapping Schwarz preconditioner for the resulting discrete system. Essentially, this reduces to finding an efficient preconditioner for the form ( ⋅ , ⋅ ) + ( div ⋅ , div ⋅ ) in the symmetric tensor space H ( div , Ω ) . The main difficulty comes from the well known complexity of building preconditioners for the div operator. We solve it by taking a decomposition similar to the Helmholz decomposition. Both additive and multiplicative preconditioners are studied, and the conditioner numbers are shown to be uniform with respect to the mesh size.