SUPG Method Research Articles

Bubble stabilization of linear finite element methods for nonlinear evolutionary convection–diffusion equations

A spatial stabilization via bubble functions of linear finite element methods for nonlinear evolutionary convection–diffusion equations is discussed. The method of lines with SUPG discretization in space leads to numerical schemes that are not only difficult to implement, when considering nonlinear evolutionary equations, but also do not produce satisfactory results. The method we propose can be seen as an alternative to this kind of methods. Once the numerical approximation belonging to a linear finite element space enriched with bubble functions is obtained, the bubble part is discarded. The linear part is shown to give a stabilized approximation to the solution being approached. The bubble functions are deduced using a linear steady convection–diffusion model in such a way that the linear part of the approximation to the linear steady model (after static condensation of bubbles) reproduces the SUPG method in the convection-dominated regime. However, for the nonlinear evolutionary equations we consider in the paper the method we propose is not equivalent to the SUPG method. Some numerical experiments are provided in the paper to show the efficiency of the procedure.

A posteriori error estimations of a SUPG method for anisotropic diffusion–convection–reaction problems

This Note presents an a posteriori residual error estimator for diffusion–convection–reaction problems approximated by a SUPG scheme on isotropic or anisotropic meshes in R d , d = 2 or 3. This estimator is based on the jump of the flux and the interior residual of the approximated solution. It is constructed to work on anisotropic meshes which account for the eventual anisotropic behavior of the solution. The equivalence between the energy norm of the error and the estimator is proved. To cite this article: T. Apel, S. Nicaise, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

COMBINING ADJOINT STABILIZED METHODS FOR THE ADVECTION-DIFFUSION-REACTION PROBLEM

Computational methods for the advection-diffusion-reaction transport equation are still a challenge. Although there exist globally stable methods, oscillations around sharp layers such as boundary, inner and outflow layers, are typical in multi-dimensional flows. In this paper a variational formulation that combines two types of stabilization integrals is proposed, namely an adjoint stabilization and a gradient adjoint stabilization. Two free parameters are chosen by imposing one-dimensional superconvergence. Then, when applied to multi-dimensional flows, the method presents better local stability than the present stabilized methods. Furthermore, in the advective-diffusive limit and for piecewise linear functional spaces, the method recovers the classical SUPG method.

Analysis of a new stabilized higher order finite element method for advection–diffusion equations

We consider a singularly perturbed advection–diffusion two-point boundary value problem whose solution has a single boundary layer. Based on piecewise polynomial approximations of degree k ⩾ 1, a new stabilized finite element method is derived in the framework of a variation multiscale approach. The method coincides with the SUPG method for k = 1 but differs from it for k ⩾ 2. Estimates for the error to an appropriate interpolant are given in several norms on different types of meshes. For k = 1 enhanced accuracy is achieved by superconvergence. Postprocessing guarantees the same estimates for the error to the solution itself.

A Unified Analysis for Conforming and Nonconforming Stabilized Finite Element Methods Using Interior Penalty

We discuss stabilized Galerkin approximations in a new framework, widening the scope from the usual dichotomy of the discontinuous Galerkin method on the one hand and Petrov--Galerkin methods such as the SUPG method on the other. The idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation, thus circumventing the need of a Petrov--Galerkin-type choice of spaces. This is made possible by adding a higher-order penalty term giving L2-control of the jumps in the gradients between adjacent elements. We consider convection-diffusion-reaction problems using piecewise linear approximations and prove optimal order a priori error estimates for two different finite element spaces, the standard H1-conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides (the Crouzeix--Raviart element). Moreover, we show how the formulation extends to discontinuous Galerkin interior penalty methods in a natural way by domain decomposition using Nitsche's method.

Stability of the SUPG finite element method for transient advection–diffusion problems

Implicit time integration coupled with SUPG discretization in space leads to additional terms that provide consistency and improve the phase accuracy for convection dominated flows. Recently, it has been suggested that for small Courant numbers these terms may dominate the streamline diffusion term, ostensibly causing destabilization of the SUPG method. While consistent with a straightforward finite element stability analysis, this contention is not supported by computational experiments and contradicts earlier Von-Neumann stability analyses of the semidiscrete SUPG equations. This prompts us to re-examine finite element stability of the fully discrete SUPG equations. A careful analysis of the additional terms reveals that, regardless of the time step size, they are always dominated by the consistent mass matrix. Consequently, SUPG cannot be destabilized for small Courant numbers. Numerical results that illustrate our conclusions are reported.

A dynamic approach for evaluating parameters in a numerical method

A new methodology for evaluating unknown parameters in a numerical method for solving a partial differential equation is developed. The main result is the identification of a functional form for the parameters which is derived by requiring the numerical method to yield ‘optimal’ solutions over a set of finite-dimensional function spaces. The functional depends upon the numerical solution, the forcing function, the set of function spaces, and the definition of the optimal solution. It does not require exact or approximate analytical solutions of the continuous problem, and is derived from an extension of the variational Germano identity. This methodology is applied to the one-dimensional, linear advection–diffusion problem to yield a non-linear dynamic diffusivity method. It is found that this method yields results that are commensurate to the SUPG method. The same methodology is then used to evaluate the Smagorinsky eddy viscosity for the large eddy simulation of the decay of homogeneous isotropic turbulence in three dimensions. In this case the resulting method is found to be more accurate than the constant-coefficient and the traditional dynamic versions of the Smagorinsky model. Copyright © 2004 John Wiley & Sons, Ltd.

A Stabilized Three-Field Formulation for Advection-Diffusion Equations

In this paper, we propose a new stabilized three-field formulation applied to the advection-diffusion equation. Using finite elements with SUPG stabilization in the interior of the subdomains our approach enables us to use almost arbitrary discrete function spaces. They need not to satisfy the inf-sup conditions of the standard three-field formulation. We can prove the stability of the scheme and an a priori estimate which is of the same convergence order as the standard SUPG method. Furthermore, the scheme can be solved efficiently by an adapted Schur complement equation. Finally some numerical experiments confirm our theoretical results.

Nearly H1-optimal finite element methods

We examine the problem of finding the H 1 projection onto a finite element space of an unknown field satisfying a specified boundary value problem. Solving the projection problem typically requires knowing the exact solution. We circumvent this issue and obtain a Petrov–Galerkin formulation which achieves H 1 optimality. Requiring weighting functions to be defined locally on the element level permits only approximate H 1 optimality in multi-dimensional configurations. We investigate the relation between our formulation and other stabilized FEM formulations. We show, in particular, that our formulation leads to a derivation of the SUPG method. In special cases, the present formulation reduces to that of residual-free bubbles. Finally, we present guidelines for obtaining the Petrov weight functions, and include a numerical example for the Helmholtz equation.

NUMERICAL SIMULATION OF TWO-DIMENSIONAL COMPRESSIBLE TURBULENT FLOWS IN EJECTORS

This paper presents a finite element method for simulation of two-dimensional internal compressible turbulent flows. The main objective of this work is to present a more adequate formulation for internal flows and to compare the performance of algebraic and one equation turbulence models for the simulation of turbulent flows in ejectors. The Reynolds-averaged Navier-Stokes equations are solved in terms of the so-called enthalpic variables: the static pressure p, the momentum per unit volume U and the total specific enthalpy h. The numerical method is a variant of the SUPG method augmented with a discontinuity capturing operator and uses a mixed finite element approximation for space discretization and a second order differentiating scheme for time discretization. For turbulence modelling, two models are considered and compared. The first one we propose is an algebraic model. The second one is the Spalart-Allmaras one equation model. Numerical simulations of flows in two-dimensional supersonic and thrust a...

Recovering SUPG using Petrov–Galerkin formulations enriched with adjoint residual-free bubbles

We consider conforming Petrov–Galerkin formulations for the advective and advective–diffusive equations. For the linear hyperbolic equation, the continuous formulation is set up using different spaces and the discretization follows with different “bubble” enrichments for the test and trial spaces. Boundary conditions for residual-free bubbles are modified to accommodate with the first-order equation case and regular bubbles are used to enrich the other space. Using piecewise linears with these enrichments, the final formulations are shown to be equivalent to the SUPG method, provided the data are assumed to be piecewise constant. Generalization to include diffusion is also presented.

Stabilization methods of bubble type for theQ1/Q1-element applied to the incompressible Navier-Stokes equations

In this paper, a general technique is developed to enlarge the velocity space of the unstable -element by adding spaces such that for the extended pair the Babuska-Brezzi condition is satisfied. Examples of stable elements which can be derived in such a way imply the stability of the well-known Q2 /Q1 -element and the 4Q1 /Q1 -element. However, our new elements are much more cheaper. In particular, we shall see that more than half of the additional degrees of freedom when switching from the Q 1 to the Q 2 and 4Q1 , respectively, element are not necessary to stabilize the Q1 /Q1 -element. Moreover, by using the technique of reduced discretizations and eliminating the additional degrees of freedom we show the relationship between enlarging the velocity space and stabilized methods. This relationship has been established for triangular elements but was not known for quadrilateral elements. As a result we derive new stabilized methods for the Stokes and Navier-Stokes equations. Finally, we show how the Brezzi-Pitkaranta stabilization and the SUPG method for the incompressible Navier-Stokes equations can be recovered as special cases of the general approach. In contrast to earlier papers we do not restrict ourselves to linearized versions of the Navier-Stokes equations but deal with the full nonlinear case.

Characteristic Galerkin method for convection-diffusion equations and implicit algorithm using precise integration

This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using precise integration method. With the operator splitting procedure, the precise integration method is introduced to determine the material derivative in the convection-diffusion equation, consequently, the physical quantities of material points. An implicit algorithm with a combination of both the precise and the traditional numerical integration procedures in time domain in the Lagrange coordinates for the characteristic Galerkin method is formulated. The stability analysis of the algorithm shows that the unconditional stability of present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results validate the presented method in solving convection-diffusion equations. As compared with SUPG method and explicit characteristic Galerkin method, the present method gives the results with higher accuracy and better stability.

Finite element implementation of two-equation and algebraic stress turbulence models for steady incompressible flows

The main purpose of this paper is to describe a finite element formulation for solving the equations for k and ε of the classical k–ε turbulence model, or any other two-equation model. The finite element discretization is based on the SUPG method together with a discontinuity capturing technique to deal with sharp internal and boundary layers. The iterative strategy consists of several nested loops, the outermost being the linearization of the Navier–Stokes equations. The basic k–ε model is used for the implementation of an algebraic stress model that is able to account for the effects of rotation. Some numerical examples are presented in order to show the performance of the proposed scheme for simulating directly steady flows, without the need of reaching the steady state through a transient evolution. Copyright © 1999 John Wiley & Sons, Ltd.

Numerical prediction of the fiber orientation in steady flows

ABSTRACT Modelling fibers orientation induced by the flow of short fiber reinforced thermoplastic involves a classical anisotropic Stokes flow problem and a hyperbolic orientation equation. This paper aims to achieve a comparison between different solution techniques suited to the hyperbolicity of the orientation equation (viz. the method of characteristics, the SUPG method and the discontinuous finite volume method).

Further considerations on residual-free bubbles for advective-diffusive equations

We further consider the Galerkin method for advective-diffusive equations in two dimensions. The finite dimensional space employed is of piecewise polynomials enriched with residual-free bubbles (RFB). We show that, in general, this method does not coincide with the SUPG method, unless the piecewise polynomials are spanned by linear functions. Furthermore, a simple stability analysis argument displays the effect of the RFB on the reduced space of piecewise polynomials, which, in some situations, is not equivalent to streamline diffusion for bilinears.

Comparison of some finite element methods for solving the diffusion-convection-reaction equation

In this paper we describe several finite element methods for solving the diffusion-convection-reaction equation. None of them is new, although the presentation is non-standard in an effort to emphasize the similarities and differences between them. In particular, it is shown that the classical SUPG method is very similar to an explicit version of the Characteristic-Galerkin method, whereas the Taylor-Galerkin method has a stabilization effect similar to a sub-grid scale model, which is in turn related to the introduction of bubble functions.

Une méthode d'éléments finis pour les écoulements internes compressibles: application aux éjecteurs

ABSTRACT This paper presents a finite element method for the simulation of two-dimensional internal compressible flows. The Navier-Stokes equations are solved in terms of the so-called enthalpic variables: the static pressure p, the momentum per unit volume U and the total specific enthalpy h. The variational formulation is a variante of the SUPG method. The stability of this method is reenforced by the use of a discontinuity capturing operator. A turbulence algebraic model for the simulation of flows in ejectors is built. It consists in separating the flow into two regions: one near the wall where the Baldwin-Lomax model is applied and the other, far from the wall, where a new formulation, based on the Schlichting model for free jets, is proposed. The turbulent viscosity evaluation, on an unstructured grid, is a variante of the Rostand's technic. The discretization of the variational form is done on a P1/P2 element and uses an implicit scheme. The algebraic system is solved using the GMRES algorithm with...

Multidimensional upwinding: Its relation to finite elements

AbstractVertex‐based multidimensional upwind schemes for scalar advection are compared with shock‐capturing SUPG finite element methods based on linear triangular elements. Both methods share the same compact stencil and are formulated as cell‐wise residual distribution methods. The distribution for the finite element method is 1/3, supplemented with a Lax‐Wendrov‐type dissipation term, while the distribution for the upwind schemes is limited to the downstream nodes of the element. The multidimensional upwind schemes use positivity as the monotonicity criterion, while the finite element method includes a residual‐based non‐linear dissipation.For hyperbolic systems such as the compressible Euler equations the upwind method relies on a multidimensional wave model to decompose the residual into scalar contributions. From this observation a new SUPG formulation for systems is proposed in which the scalar SUPG method is applied to each of the decomposed residuals obtained from the wave model, thereby providing a better‐founded definition of the τ dissipation matrix and shock‐capturing term in the SUPG methods.

Une méthode d'éléments finis pour le calcul des écoulements compressibles utilisant les variables conservatives et la méthode SUPG

ABSTRACT The Navier-Stokes and Euler equations are solved in a conservative form and using the conservation variables. The variational formulation and the corresponding finite element approximations are discussed. For high speed flows, the numerical model has to be stabilized. We used a SUPG method along with a shock capturing operator. The stabilization techniques and the solution algorithm are described. Some numerical tests are carried out to validate the code. This model is employed to simulate the flow in an engine intake.