Discrete Inf Sup Research Articles

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

AbstractA 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.

LOWER ORDER RECTANGULAR NONCONFORMING MIXED FINITE ELEMENT FOR THE THREE-DIMENSIONAL ELASTICITY PROBLEM

In this paper, we propose a first-order rectangular nonconforming element for the stress-displacement system derived from the Hellinger–Reissner variational principle for the three-dimensional elasticity problem. We show that the discrete inf–sup condition holds for this scheme. Based on some superconvergence of the consistency error, we prove the optimal error estimate of [Formula: see text] for both the displacement and stress.

Penalty finite element approximations for the Stokes equations by L2 projection

AbstractThis paper considers the penalty finite element method for the Stokes equations, based on some stable finite elements space pair (Xh, Mh) that do satisfy the discrete inf–sup condition. Theoretical results show that the penalty error converges as fast as one should expect from the order of the elements. Moreover, the penalty finite element method by L2 projection can improve the penalty error estimates. Finally, we confirm these results by a series of numerical experiments. Copyright © 2008 John Wiley & Sons, Ltd.

Penalty finite element method for Stokes problem with nonlinear slip boundary conditions

The penalty finite element method for Stokes problem with nonlinear slip boundary conditions, based on the finite element subspace ( V h , M h ) which satisfies the discrete inf–sup condition, is investigated in this paper. Since this class of nonlinear slip boundary conditions include the subdifferential property, the weak variational formulation associated with Stokes problem is variational inequality. Under some regularity assumptions, we obtain the optimal H 1 and L 2 error estimates between u and u h , and between u and u h ε , where the error orders are ε + h for H 1 error and ε + h 2 for L 2 error.

Some preconditioners for the CFIE equation of electromagnetism

AbstractWe present three weak parametrices of the operator of the combined field integral equation (CFIE). An interesting feature of these parametrices is that they are compatible with different discretization strategies and hence allow for the construction of efficient preconditioners dedicated to the CFIE. Their numerical analysis shows that a regularization process acting at the continuous level of the equation is also effective at the discrete level if the mesh size tends to zero. First numerical tests confirm this effect and preconditioning is observed indeed. Furthermore, we show that the underlying operator of CFIE satisfies a uniform discrete Inf–Sup condition that allows one to predict an original convergence result for the numerical solution of CFIE to the exact one. Copyright © 2008 John Wiley & Sons, Ltd.

Stabilized finite element schemes for incompressible flow using Scott–Vogelius elements

We propose a stabilized mixed finite element method based on the Scott–Vogelius element for the Oseen equation. Here, only convection has to be stabilized since by construction both the discrete pressure and the divergence of the discrete velocities are controlled in the norm L 2 . As stabilization we propose either the local projection stabilization or the interior penalty stabilization based on the penalization of the gradient jumps over element edges. We prove a discrete inf–sup condition leading to optimal a priori error estimates. Moreover, convergence of the velocities is completely independent of the pressure regularity, and in the purely incompressible case the discrete velocities are pointwise divergence free. The theoretical considerations are illustrated by some numerical examples.

A unified convergence analysis for local projection stabilisations applied to the Oseen problem

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.

Topology optimization of incompressible media using mixed finite elements

We present an alternative formulation for the topology optimization of structures made of incompressible materials, a topic that cannot be tackled using most of the approaches of the current literature that are mainly based on displacement finite elements that are well known to be affected by the locking phenomenon. A way out of the problem has recently been proposed in Sigmund and Clausen (2007) [O. Sigmund, P.M. Clausen, Topology optimization using a mixed formulation: an alternative way to solve pressure load problems, Comput. Methods Appl. Mech. Engrg. 196 (13–16) (2007) 1874–1889], based on a displacement–pressure finite element discretization. The approach presented here consists in a truly-mixed variational formulation coupled to a mixed-FEM discretization that uses the composite element of Johnson and Mercier for the discretization of the stress field. By so doing, the continuous and discrete inf–sup conditions of the problem are automatically met even in the presence of incompressible materials. A few numerical studies are presented to validate the theoretical framework for which the well-known method of moving asymptotes (MMA) is adopted for the numerical optimization of the problem. Different topologies in plane stress and plane strain conditions are presented, with particular attention to the convergence of the last ones to pure 0–1 designs. Some forthcoming investigations are eventually highlighted including the solution of stress-constrained topology-optimization problems that find in the truly-mixed setting their natural environment and the extension of the present formulation to deal with pressure–load structural problems.

Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem

We consider the Stokes problem in an axisymmetric three-dimensional domain with data which are axisymmetric and have angular component equal to zero. We observe that the solution is also axisymmetric and the velocity has also zero angular component, hence the solution satisfies a system of equations in the meridian domain. The weak three-dimensional problem reduces to a two-dimensional one with weighted integrals. The latter is discretized by Taylor–Hood type finite elements. A weighted Clément operator is defined and approximation results are proved. This operator is then used to derive the discrete inf–sup condition and optimal a priori error estimates.

A pressure-Poisson stabilized finite element method for the non-stationary Stokes equations to circumvent the inf–sup condition

In this article, stabilized finite element methods are considered for the non-stationary Stokes equations, based on some lowest equal-order finite elements space pair ( X h , M h ) which do not satisfy the discrete inf–sup condition. The stability of two kinds of methods is derived under some regularity assumptions. Then, the convergence of the penalty method and the pressure-Poisson stabilized method is compared. The result shows that the former error limits the order of approximation to O ( ϵ + h / ϵ ) , and the latter yields the optimal error estimate O( h).

Stabilized hp-DGFEM for Incompressible Flow

We consider stabilized mixed hp-discontinuous Galerkin methods for the discretization of the Stokes problem in three-dimensional polyhedral domains. The methods are stabilized with a term penalizing the pressure jumps. For this approach it is shown that ℚk-ℚk and ℚk-ℚk-1 elements satisfy a generalized inf–sup condition on geometric edge and boundary layer meshes that are refined anisotropically and non quasi-uniformly towards faces, edges, and corners. The discrete inf–sup constant is proven to be independent of the aspect ratios of the anisotropic elements and to decrease as k-1/2 with the approximation order. We also show that the generalized inf–sup condition leads to a global stability result in a suitable energy norm.

A quasi-local interpolation operator�preserving the discrete divergence

We construct an operator that preserves the discrete divergence and has the same quasi-local approximation properties as a regularizing interpolant; this is very useful when discretizing nonlinear incompressible fluid models. For low-degree finite elements, such operators have an explicit expression, from which local approximation properties can be easily derived. But for higher-degree finite elements, an explicit expression is generally not available and this construction is achieved by proving a global discrete inf–sup condition while using only local arguments. We write this construction in a general case, for conforming and non-conforming elements, and then give some applications.

On the quadrilateral Q2–P1 element for the Stokes problem

AbstractThe 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.

Computational studies on generation and control in magnetron sputtering plasma

We devise and analyze vertex-based schemes on polyhedral meshes to approximate advection–reaction equations. Error estimates of order O(h3/2) are established in the discrete inf–sup stability norm which includes the mesh-dependent weighted advective derivative. The two key ingredients are a local polyhedral reconstruction map leaving affine polynomials invariant, and a local design of stabilization whereby gradient jumps are only penalized across some subfaces in the interior of each mesh cell. Numerical results are presented on three-dimensional polyhedral meshes.

Mixed hp finite element methods for Stokes and non-Newtonian flow

We analyze the stability of hp finite elements for viscous incompressible flow. For the classical velocity—pressure formulation, we give new estimates for the discrete inf—sup constants on geometric meshes which are explicit in the polynomial degree k of the elements. In particular, we obtain new bounds for p-elements on triangles. For the three-field Stokes problem describing linearized non-Newtonian flow, we estimate discrete inf—sup constants explicit in both h and k for various subspace choices (continuous and discontinuous) for the extra-stress. We also give a stability analysis of the hp-version of an elastic-viscous-split-stress (EVSS) method and present elements that are stable and optimal in h and k. Finally, we present numerical results that show the exponential convergence of the hp version for Stokes flow over unsmooth domains.

MIXED hp-FEM ON ANISOTROPIC MESHES

Mixed hp-FEM for incompressible fluid flow on anisotropic meshes are analyzed. A discrete inf–sup condition is proved with a constant independent of the meshwidth and the aspect ratio. For each polynomial degree k≥2 we present velocity-pressure subspace pairs which are stable on quadrilateral mesh-patches independently of the element aspect ratio, implying in particular divergence stability on the so-called Shishkin-meshes. Moreover, the inf–sup constant is shown to depend on the spectral order k like k-1/2 for quadrilateral meshes and like k-3 for meshes containing triangles. New consistency results for spectral elements on anisotropic meshes are also proved.

Least-squares finite element for second order boundary value problem with optimal rates of convergence

During the past twenty years, engineers and mathematicians have successfully developed the mixed methods and the hybrid methods to solve problems (1.1), (1.2) and (1.3) numerically, such as [1–6]. The discrete inf-sup or its equivalent condition for the subspaces of the approximate functions of u and p has been applied for. It ensures the numerical results converge to the real world solution steadily. In this paper, we try to put an extra equation which is the compatibility condition of this system. We then apply the least-squares method to the so-called enriched system. Analysis shows that this method achieves the optimal rate of convergence in the H 1 and L 2 norms without using the above condition. Numerical examples are still available.