Stabilized finite element methods: I. Application to the advective-diffusive model

Franca, Leopoldo P.; Frey, Sérgio; Hughes, Thomas J.R.

doi:10.1016/0045-7825(92)90143-8

Cited by 592 publications

(398 citation statements)

References 12 publications

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“…Finally we recall that the pseudo bubble functions B * i (i = 1, 2) are approximations to B i on the sub-grid specified above, through (12) and they are used in place of B i to represent u B in (11). The approximate representation of u B by bubble functions B * i (i = 1, 2) is eventually used to solve (7) for its linear part.…”

Section: Remarkmentioning

confidence: 99%

“…The plain Galerkin method may not work for such problems on reasonable discretizations, producing unphysical oscillations. The SUPG method, and its variants, are among the most popular approaches to overcome that difficulty, which are based on augmenting the variational formulation by mesh-dependent terms in order to gain control over the derivatives of the solution [10,12,16]. The great advantage of this approach is not only its generality, but also its error analysis can be performed in many cases of interest.…”

Section: Introductionmentioning

confidence: 99%

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Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems

Sendur

Neslitürk

2011

Calcolo

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It is known that the enrichment of the polynomial finite element space of degree 1 by bubble functions results in a stabilized scheme of the SUPG-type for the convection-diffusion-reaction problems. In particular, the residual-free bubbles (RFB) can assure stabilized methods, but they are usually difficult to compute, unless the configuration is simple. Therefore it is important to devise numerical algorithms that provide cheap approximations to the RFB functions, contributing a good stabilizing effect to the numerical method overall. Here we propose a stabilization technique based on the RFB method and particularly designed to treat the most interesting case of small diffusion. We replace the RFB functions by their cheap, yet efficient approximations which retain the same qualitative behavior. The approximate bubbles are computed on a suitable sub-grid, the choice of whose nodes are critical and determined by minimizing the residual of a local problem with respect to L 1 norm. The resulting numerical method has similar stability features with the RFB method for the whole range of problem parameters. This fact is also confirmed by numerical experiments. We also note that the location of the sub-grid nodes suggested by the strategy herein coincides with the one in Brezzi et al. (Math. Models Methods Appl. Sci. 13:445-461, 2003).

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Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems

Sendur

Neslitürk

2011

Calcolo

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“…Equation (1) represents the balance of linear momentum, and equation (2) represents the continuity equation for an incompressible continuum. Equations (3) and (4) are the Dirichlet and Neumann boundary conditions, respectively.…”

Section: Governing Equations For the Stokes Problemmentioning

confidence: 99%

“…By stabilized finite element methods we mean methods that add mesh dependent terms to the standard Galerkin formulation that enable the formulation to satisfy or circumvent the LBB condition [2]. In contrast, enriched finite element methods add bubble functions to the finite element function space, which in turn play a stabilizing role.…”

Section: Introductionmentioning

confidence: 99%

On the stability of bubble functions and a stabilized mixed finite element formulation for the Stokes problem

Turner

Nakshatrala

Hjelmstad

2008

Numerical Methods in Fluids

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Abstract. In this paper we investigate the relationship between stabilized and enriched finite element formulations for the Stokes problem. We also present a new stabilized mixed formulation for which the stability parameter is derived purely by the method of weighted residuals. This new formulation allows equal order interpolation for the velocity and pressure fields. Finally, we show by counterexample that a direct equivalence between subgrid-based stabilized finite element methods and Galerkin methods enriched by bubble functions cannot be constructed for quadrilateral and hexahedral elements using standard bubble functions.

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“…Unlike the classical stabilized finite element methods that are often advocated for the finite element solution of advection-diffusion problems in the high Péclet regime, e.g., the streamline upwind PetrovGalerkin (SUPG) method [11,27], Galerkin least-squares (GLS) method [25,28], and the unusual stabilized finite element method (USFEM) [19,20], the main idea of DEM is to enrich the standard piecewise polynomial approximations by non-conforming and non-polynomial basis functions that are related to the partial differential equation (PDE) to be solved. In DEM, these functions are chosen as the free-space solutions of the homogeneous constant-coefficient counterpart of the governing PDE.…”

Section: Introductionmentioning

confidence: 99%

A higher‐order discontinuous enrichment method for the solution of high péclet advection–diffusion problems on unstructured meshes

Farhat

Kalashnikova

Tezaur

2009

Numerical Meth Engineering

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SUMMARYA higher-order discontinuous enrichment method (DEM) with Lagrange multipliers is proposed for the efficient finite element solution on unstructured meshes of the advection-diffusion equation in the high Péclet number regime. Following the basic DEM methodology, the usual Galerkin polynomial approximation is enriched with free-space solutions of the governing homogeneous partial differential equation (PDE). In this case, these are exponential functions that exhibit a steep gradient in a specific flow direction. Exponential Lagrange multipliers are introduced at the element interfaces to weakly enforce the continuity of the solution. The construction of several higher-order DEM elements fitting this paradigm is discussed in details. Numerical tests performed for several two-dimensional benchmark problems demonstrate their computational superiority over stabilized Galerkin counterparts, especially for high Péclet numbers.

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Stabilized finite element methods: I. Application to the advective-diffusive model

Cited by 592 publications

References 12 publications

Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems

Applications of the pseudo residual-free bubbles to the stabilization of convection-diffusion-reaction problems

On the stability of bubble functions and a stabilized mixed finite element formulation for the Stokes problem

A higher‐order discontinuous enrichment method for the solution of high péclet advection–diffusion problems on unstructured meshes

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