Galerkin Least-squares Research Articles

A remark on least-squares Galerkin procedures for pseudohyperbolic equations

In this paper, we introduce two split least-squares Galerkin finite element procedures for pseudohyperbolic equations arising in the modelling of nerve conduction process. By selecting the least-squares functional properly, the procedures can be split into two sub-procedures, one of which is for the primitive unknown variable and the other is for the flux. The convergence analysis shows that both the two methods yield the approximate solutions with optimal accuracy in L 2 ( Ω ) norm for u and u t and ( L 2 ( Ω ) ) 2 norm for the flux σ . Moreover, the two methods get approximate solutions with first-order and second-order accuracy in time increment, respectively. A numerical example is given to show the efficiency of the introduced schemes.

A priori and a posteriori error analysis of an augmented mixed finite element method for incompressible fluid flows

In this paper we extend recent results on the a priori and a posteriori error analysis of an augmented mixed finite element method for the linear elasticity problem, to the case of incompressible fluid flows with symmetric stress tensor. Similarly as before, the present approach is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and from the relations defining the pressure in terms of the stress tensor and the rotation in terms of the displacement, all of them multiplied by stabilization parameters. We show that these parameters can be suitably chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well-posed for any choice of finite element subspaces. Next, we present a reliable and efficient residual-based a posteriori error estimator for the augmented mixed finite element scheme. Finally, several numerical results confirming the theoretical properties of this estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities and the large stress regions of the solution, are reported.

Editorial

Editorial

Galerkin Least-Squares Multifield Approximations for Flows of Inelastic Non-Newtonian Fluids

The aim of this work is to investigate a Galerkin least-squares (GLS) multifield formulation for inelastic non-Newtonian fluid flows. We present the mechanical modeling of isochoric flows combining mass and momentum balance laws in continuum mechanics with an inelastic constitutive equation for the stress tensor. For the latter, we use the generalized Newtonian liquid model, which may predict either shear-thinning or shear-thickening. We employ a finite element formulation stabilized via a GLS scheme in three primal variables: extra stress, velocity, and pressure. This formulation keeps the inertial terms and has the capability of predicting viscosity dependency on the strain rate. The GLS method circumvents the compatibility conditions that arise in mixed formulations between the approximation functions of pressure and velocity and, in the multifield case, of extra stress and velocity. The GLS terms are added elementwise, as functions of the grid Reynolds number, so as to add artificial diffusivity selectively to diffusion and advection dominant flow regions—an important feature in the case of variable viscosity fluids. We present numerical results for the lid-driven cavity flow of shear-thinning and shear-thickening fluids, using the power-law viscosity function for Reynolds numbers between 50 and 500 and power-law exponents from 0.25 to 1.5. We also present results concerning flows of shear-thinning Carreau fluids through abrupt planar and axisymmetric contractions. We study ranges of Carreau numbers from 1 to 100, Reynolds numbers from 1 to 100, and power-law exponents equal to 0.1 and 0.5. Besides accounting for inertia effects in the flow, the GLS method captures some interesting features of shear-thinning flows, such as the reduction of the fluid stresses, the flattening of the velocity profile in the contraction plane, and the separation of the boundary layer downstream the contraction.

On a multi-scale element-free Galerkin method for the Stokes problem

In this paper, a multi-scale element-free Galerkin method is presented for the Stokes problem. The new method is based on the Hughes’ variational multi-scale formulation, and arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolved scales. In this method, an unresolved model is obtained in which unresolved scales are incorporated analytically through the bubble functions. Modeling of the unresolved scales corrects the lack of the stability of the standard element-free Galerkin formulation and the resulting stabilized formulation possesses superior properties like that of the streamline upwind/Petrov–Galerkin (SUPG) method and the Galerkin/least-squares (GLS) method. The method allows equal order basis for pressure and velocity because it violates the celebrated Babuska–Brezzi condition. A significant feature of the present method is that the structure of the stabilization tensor τ appears naturally via the solution of the fine-scale problem. Numerical results for example problems confirm that this method has some excellent properties, such as better stability and accuracy.

Galerkin least-squares solutions for purely viscous flows of shear-thinning fluids and regularized yield stress fluids

This paper aims to present Galerkin Least-Squares approximations for flows of Bingham plastic fluids. These fluids are modeled using the Generalized Newtonian Liquid (GNL) constitutive equation. Their viscoplastic behavior is predicted by the viscosity function, which employs the Papanastasiou's regularization in order to predict a highly viscous behavior when the applied stress lies under the material's yield stress. The mechanical modeling for this type of flow is based on the conservation equations of mass and momentum, coupled to the GNL constitutive equation for the extra-stress tensor. The finite element methodology concerned herein, the well-known Galerkin Least-Squares (GLS) method, overcomes the two greatest Galerkin shortcomings for mixed problems. There is no need to satisfy Babuška-Brezzi condition for velocity and pressure subspaces, and spurious numerical oscillations, due to the asymmetric nature of advective operator, are eliminated. Some numerical simulations are presented: first, the lid-driven cavity flow of shear-thinning and shear-thickening fluids, for the purpose of code validation; second, the flow of shear-thinning fluids with no yield stress limit, and finally, Bingham plastic creeping flows through 2:1 planar and axisymmetric expansions, for Bingham numbers between 0.2 and 133. The numerical results illustrate the arising of two distinct unyielded regions: one near the expansion corner and another along the flow centerline. For those regions, velocity and pressure fields are investigated for the various Bingham numbers tested.

A posteriori error estimates for mixed finite element approximations of elliptic problems

We derive residual based a posteriori error estimates of the flux in L 2-norm for a general class of mixed methods for elliptic problems. The estimate is applicable to standard mixed methods such as the Raviart–Thomas–Nedelec and Brezzi–Douglas–Marini elements, as well as stabilized methods such as the Galerkin-Least squares method. The element residual in the estimate employs an elementwise computable postprocessed approximation of the displacement which gives optimal order.

On finite element formulations for nearly incompressible linear elasticity

In this paper we present a mixed stabilized finite element formulation that does not lock and also does not exhibit unphysical oscillations near the incompressible limit. The new mixed formulation is based on a multiscale variational principle and is presented in two different forms. In the first form the displacement field is decomposed into two scales, coarse-scale and fine-scale, and the fine-scale variables are eliminated at the element level by the static condensation technique. The second form is obtained by simplifying the first form, and eliminating the fine-scale variables analytically yet retaining their effect that results with additional (stabilization) terms. We also derive, in a consistent manner, an expression for the stabilization parameter. This derivation also proves the equivalence between the classical mixed formulation with bubbles and the Galerkin least-squares type formulations for the equations of linear elasticity. We also compare the performance of this new mixed stabilized formulation with other popular finite element formulations by performing numerical simulations on three well known test problems.

An algebraic subgrid scale finite element method for the convected Helmholtz equation in two dimensions with applications in aeroacoustics

An algebraic subgrid scale finite element method formally equivalent to the Galerkin Least-Squares method is presented to improve the accuracy of the Galerkin finite element solution to the two-dimensional convected Helmholtz equation. A stabilizing term has been added to the discrete weak formulation containing a stabilization parameter whose value turns to be the key for the good performance of the method. An appropriate value for this parameter has been obtained by means of a dispersion analysis. As an application, we have considered the case of aerodynamic sound radiated by incompressible flow past a two-dimensional cylinder. Following Lighthill’s acoustic analogy, we have used the time Fourier transform of the double divergence of the Reynolds stress tensor as a source term for the Helmholtz and convected Helmholtz equations and showed the benefits of using the subgrid scale stabilization.

A Priori and a Posteriori Error Analysis of a Wavelet-Based Stabilization for the Mixed Finite Element Method

We use Galerkin least-squares terms and biorthogonal wavelet bases to develop a new stabilized dual-mixed finite element method for second-order elliptic equations in divergence form with Neumann boundary conditions. The approach introduces the trace of the solution on the boundary as a new unknown that acts also as a Lagrange multiplier. We show that the resulting stabilized dual-mixed variational formulation and the associated discrete scheme defined with Raviart–Thomas spaces are well-posed and derive the usual a priori error estimates and the corresponding rate of convergence. Furthermore, a reliable and efficient residual-based a posteriori error estimator and a reliable and quasi-efficient one are provided.

CONTINUOUS Q1–Q1 STOKES ELEMENTS STABILIZED WITH NON-CONFORMING NULL EDGE AVERAGE VELOCITY FUNCTIONS

We present a stabilized finite element method for Stokes equations with piecewise continuous bilinear approximations for both velocity and pressure variables. The velocity field is enriched with piecewise polynomial bubble functions with null average at element edges. These functions are statically condensed at the element level and therefore they can be viewed as a continuous Q1–Q1 stabilized finite element method. The enriched velocity-pressure pair satisfies optimal inf–sup conditions and approximation properties. Numerical experiments show that the proposed discretization outperforms the Galerkin least-squares method.

Construction of stabilization operators for Galerkin least-squares discretizations of compressible and incompressible flows

The design and analysis of a class of stabilization operators suitable for space–time Galerkin least-squares finite element discretizations of the symmetrized compressible Navier–Stokes equations is discussed. The obtained stabilization matrix is well defined in the incompressible limit and reduces to the matrix described in [M. Polner, J.J.W. van der Vegt, R.M.J. van Damme, Analysis of stabilization operators for Galerkin least-squares discretizations of the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg. 195 (2006) 982–1006]. The stabilization matrix is investigated to give necessary and sufficient conditions for its positive definiteness. Under these conditions on the stabilization matrix, the Galerkin least-squares method for the symmetrized compressible Navier–Stokes equations satisfies the entropy condition, which ensures global nonlinear stability of the discretization.

Least-squares Galerkin procedures for pseudohyperbolic equations

Two least-squares Galerkin finite element schemes are formulated to solve pseudohyperbolic equations. The advantage of this method is that it is not subject to the LBB condition. The convergence analysis shows that the methods both yield the approximate solutions with optimal accuracy in ( L 2 ( Ω ) ) 2 × L 2 ( Ω ) . Moreover, the two methods get the approximate solutions with first-order and second-order accuracy in time increment, respectively. Numerical example shows that the two schemes are effective.

Coupling Stokes and Darcy equations

We study an interface problem between a fluid flow, governed by Stokes equations, and a flow in a porous medium, governed by Darcy equations. We consider a weak formulation of the coupled problem which allows to use classical Stokes finite elements in the fluid domain, and standard continuous piecewise polynomials in the porous medium domain. Meshes do not need to match at the interface. The formulation of Stokes equations is standard while a Galerkin least-squares formulation is used for a mixed form of Darcy equations. We prove the well-posedness of the coupled problem for this formulation and the convergence for some finite element approximations. A two-dimensional numerical example is also given.

移動境界を有する浅水長波流れ解析のためのバックグラウンドメッシュに基づくメッシュ再構築手法

This paper presents a mesh re-generation technique based on background mesh for shallow water flows with moving boundaries. The stabilized space-time finite element method based on GLS (Galerkin Least Square) method is introduced for both space and time discretization to compute the moving boundary problems in shallow water flows, accurately. In order to improve the robustness of computation and the consideration of accurate landform the re-generation technique using a back-ground mesh is introduced. The present method is applied to several wave run-up problems. The efficiency of the present method is shown by numerical results.

GALERKIN LEAST-SQUARES APPROXIMATIONS FOR FLOWS OF CASSON FLUIDS THROUGH AN EXPANSION

Among non-Newtonian fluid models, purely viscous constitutive equations play an important role in industrial applications regardless their lack of accuracy in non-viscometric flows. In this work we are concerned with the flow of viscoplastic shear-thinning fluids in complex geometry. Viscoplastic fluids are those that behave as extremely high viscosity materials when submitted to low stresses and that flow when submitted to stresses higher than a yield stress value. Usually, they also present shearthinning behavior. Fluids such as molten chocolate, xanthan gum solutions, blood, wastewater sludges, muds, and polymer solutions present viscoplastic shear-thinning features. In order to approximate numerically viscoplastic shear-thinning flows we first describe a mechanical model based on continuum mechanics conservation laws of mass and momentum. The description of material behavior is such as to respect certain principles of objectivity and generality in continuum mechanics. The Generalized Newtonian Liquid constitutive equation with Casson viscosity function is able to predict viscoplasticity and shear-thinning. The numerical approximation of the equations is performed by a finite element method. To prevent the model from pathologies known for the classic Galerkin method, we employ a stabilized method based on a Galerkin least-squares (GLS) scheme, which is designed to circumvent Babuška-Brezzi condition and deal with the asymmetry of the advective operator. We present approximations for the flow through a planar 4:1 sudden expansion. We investigate the influence of Reynolds and Casson numbers on the flow dynamics.

Forchheimer law derived by homogenization of gas flow in turbomachines

A general formulation of the homogenization problem of compressible fluid flow through a periodic porous material in turbomachines is presented here. This formulation is able to derive a Forchheimer law with a mean velocity dependent permeability as equivalent macroscopic behavior. To specify this permeability, additional flow problems are defined on the unit cell and solved by a mixed stabilized finite element discretization. The application of the Galerkin least-square (GLS) method requires the introduction of two stabilization terms with appropriate parameters. The mixed finite element discretization of these unit cell problems is finally outlined.

An augmented discontinuous Galerkin method for elliptic problems

In this Note we propose an augmented discontinuous Galerkin method for elliptic linear problems in the plane with mixed boundary conditions. Our approach introduces Galerkin least-squares terms, arising from constitutive and equilibrium equations, which allow us to look for the flux unknown in the local Raviart–Thomas space. The unique solvability is established avoiding the introduction of lifting operators and a Céa estimate is derived, which yields the rate of convergence of error, measured in an appropriate norm, being optimal respect to the h-version. We emphasize that for practical computations, this method reduces the degrees of freedom, with respect to the classical discontinuous Galerkin method. To cite this article: T.P. Barrios, R. Bustinza, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Stabilized 3D finite elements for the numerical solution of the Navier–Stokes equations in semiconductors

In this article, we deal with the three-dimensional numerical simulation of semiconductor devices using the Viscous-Hydrodynamic (VHD) transport model. A reformulation of the VHD system using entropy variables allows to end up with a quasi-linear form that is symmetric and for which a stability result (in form of Clausius–Duhem inequality) is proved to hold. The numerical approximation of the VHD model is then performed using a Time-Discontinuous Galerkin Least-Squares finite element formulation including a discontinuity shock-capturing operator and based on a fully unstructured tetrahedral decomposition of the device domain. The approach combines in a unified framework the stability and optimality features of the standard Galerkin method with the ability of the scheme in effectively coping with the strong variations attained by the solution throughout the semiconductor device, as is demonstrated by numerical results in the simulation of several benchmark problems subject to quite different boundary conditions.

Four-field Galerkin/least-squares formulation for viscoelastic fluids

A new Galerkin/Least-Squares (GLS) stabilized finite element method is presented for computing viscoelastic flows of complex fluids described by the conformation tensor; it extends the well-established GLS method for computing flows of incompressible Newtonian fluids. GLS methods are attractive for large-scale computations because they yield linear systems that can be solved easily with iterative solvers (e.g., the Generalized Minimum Residual method) and because they allow simple combinations of interpolation functions that can be conveniently and efficiently implemented on modern distributed-memory cache-based clusters. Like other state-of-the-art methods for computing viscoelastic flows (e.g., DEVSS-TG/SUPG), the new GLS method introduces a separate variable to represent the velocity gradient; with the aid of this variable, the conservation equations of mass, momentum, conformation, and the definition of velocity gradient are converted into a set of first-order partial differential equations in four unknown fields—pressure, velocity, conformation, and velocity gradient. The unknown fields are represented by low-order (continuous piecewise linear or bilinear) finite element basis functions. The method is applied to the Oldroyd-B constitutive equation and is tested in two benchmark problems—flow in a planar channel and flow past a cylinder in a channel. Results show that (1) the mesh-convergence rate of GLS is comparable to the DEVSS-TG/SUPG method; (2) the LS stabilization permits using equal-order basis functions for all fields; (3) GLS handles effectively the advective terms in the evolution equation of the conformation tensor; and (4) GLS yields accurate results at lower computational costs than DEVSS-type methods.