Galerkin formulation

The weight function used in the Galerkin formulation can be identical to either of the shape functions of a two-node linear element, therefore, for each weight function an equation corresponding to the weak statement (2.53) is derived... 

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. 

The Galerkin formulation results in a large set of differential/algebraic equations that are most easily expressed in the form... 

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

In order to solve viscoelastic problems, we must select the most convenient model for the stress and then proceed to develop the finite element formulation. Doue to the excess in non-linearity and coupling of the viscoelastic momentum equations, three distinct Galerkin formulations are used for the governing equations, i.e., we use different shape functions for the viscoelastic stress, the velocity and the pressure... 

The Lesaint-Raviart (discontinuous Galerkin) method [62] is based on discontinuous approximations of the extra-stress components and is characterized by an element per element treatment of the constitutive equation. This leads to the introduction of a stress step in the Galerkin formulation, from the preceding... 

In the early 1970s, the standard finite element approximations were based upon the Galerkin formulation of the method of weighted residuals. This technique did emerge as a powerful numerical procedure for solving elliptic boundary value problems [102, 75, 53, 84, 50, 89, 17, 35]. The Galerkin finite element methods are preferable for solving Laplace-, Poisson- and and diffusion equations because they do not require that a variational principle exists for the problem to be analyzed. However, the power of the method is still best utilized in systems for which a variational principle exists, and it... 

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. 

The most important full CLD methods are the extended methods of moments, such as the probabihty generating function method and coarse-graining-based techniques the fixed pivot method and the discrete weighted Galerkin formulation. In what follows, the most important aspects of these methods are highlighted. 

In the discrete weighted Galerkin formulation, the CLD is represented by an expansion in basis functions until a truncated value (A/tr) (Budde and Wulkow, 1991 ledema et al., 2003 Wulkow, 1996). For example, for the dead polymer this implies that the CLD at time t is... 

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 ... 

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