Petrov-Galerkin formulation

Hughes, T. J. R., Franca, L. P. and Balestra, M., 1986. A new finite-element formulation for computational fluid dynamics. 5. Circumventing the Babuska-Brezzi condition - a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal order interpolations. Cornput. Methods Appl. Meek Eng. 59, 85-99. 

Retaining all of the terms in the w eight function a least-squares scheme corresponding to a second-order Petrov-Galerkin formulation will be obtained. 

Streamline upwind Petrov/Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations,... 

Hughes, T. J. R. Brooks, A. N. (1982) Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations. Computer Methods in Applied Mechanics and Engineering il, 199-259. 

Felicelli et al. [54] used a bilinear Lagrangian isoparametric element to discretize the transport equations. The convective terms are dealt with using a Petrov-Galerkin formulation in which the weighting function is perturbed in the convective term. The perturbed weighting function is expressed as ... 

In the mathematical literature, the Galerkin method is also known as Galerkin-Bubnov, while the case Wj / Petrov-Galerkin [30,68] and is used in special finite element formulations, such as those where the heat transfer is governed by convective effects. The application of Galerkin s method in the finite element method will be covered in detail in Chapter 9 of this textbook. 

An element for the stress components composed of 16 sub-elements (4x4) on which bilinear (continuous) polynomials are used, was introduced by Marchal and Crochet in [28]. This leads to a continuous C° approximation of the three variables. The velocity is approximated by biquadratic polynomials while the pressure is linear. Fortin and Pierre ([17]) made a mathematical analysis of the Stokes problem for this three-field formulation. They conclude that the polynomial approximations of the different variables should satisfy the generalized inf-sup (Brezzi-Babuska) condition introduced by Marchal and Crochet and they proved it was the case for the Marchal and Crochet element. In order to take into account the hyperbolic character of the constitutive equation, Marchal and Crochet have implemented and compared two different methods. The first is the Streamline-Upwind/Petrov-Galerkin (SUPG). Thus a so-called non-consistent Streamline-Upwind (SU) is also considered (already used in [13]). As a test problem, they selected the "stick-slip" flow. With SUPG method applied to this problem, wiggles in the stress and the velocity field were obtained. In the SU method, the modified weighting function only applies to the convective terms in the constitutive equations. 

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