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We propose a family of mimetic discretization schemes for elliptic problems including convection and reaction terms. Our approach is an extension of the mimetic methodology for purely diffusive problems on unstructured polygonal and polyhedral meshes. The a priori error analysis relies on the connection between the mimetic formulation and the lowest order… (More)

Residual-free bubbles have been recently introduced in order to compute optimal values for the stabilization methods a la Hughes-Franca. However, unless in very special situations, (one-dimensional problems, limit cases, etc.) they require the actual solution of PDE problems (the bubble problems) in each element. Thus they are very diicult to be used in… (More)

- F Brezzi, L P Franca, A Russo
- 2007

We further consider the Galerkin method for advective-diiusive equations in two-dimensions. The nite 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… (More)

We show that three well-known \variational crimes" in nite elements { upwinding, mass lumping and selective reduced integration { may be derived from the Galerkin method employing the standard polynomial-based nite element spaces enriched with residual-free bubbles.

- F Brezzi Yx, T J R Hughes, L D Marini, A Russo, E Ssli
- 1999

We develop an a priori error analysis of a nite element approximation to the elliptic advection-diiusion equation ?"u + a ru = f subject to a homogeneous Dirichlet boundary condition, based on the use of residual-free bubble functions. An optimal order error bound is derived in the so-called stability-norm "krvk 2 L 2 (() + X T h T ka rvk 2 L 2 (T) ! 1=2 ;… (More)

- F Brezzi, L P Franca, T J R Hughes, A Russo
- 1996

We present an overview of stabilized nite element methods and of the standard Galerkin method enriched with residual-free bubble functions. The inadequacy of the standard Galerkin method using piecewise polyno-mials is discussed for diierent applications; the treatment using stabilized methods in their diierent versions is reviewed; and the connection to… (More)

- L P Franca, A Russo
- 1996

Linear-constant velocity-pressure elements are enriched with residual-free macro bubbles. Static condensation prompts a stabilized method for this element, where mesh-dependent jumps of the normal stress are added across internal edges of the underlying macroelements. This procedure renders the SIMPLEST element stable.

- Leopoldo P Franca, Alessandro Russo, L P Franca, A Russo Preprint
- 1995

Residual-free bubbles are derived for the Timoshenko beam problem. Eliminating these bubbles the resulting formulation is form-identical to using the following tricks to the standard variational formulation: i) one-point reduced integration on the shear energy term; ii) replace its coeecient 1== 2 by 1=(2 + (h 2 K =12)) in each element; iii) modify… (More)

- F Brezzi, L P Franca, D Marini, A Russo
- 1997

A nodally exact scheme is derived for a model equation in 1D involving zeroth and second order terms. The method is derived using residual-free bubbles in conjunction with the Galerkin approximation. It is shown that this approach leads to the mass lumping scheme for suciently small mesh sizes. 1. The residual-free bubbles approach. The usage of the… (More)