Discrete weighted Galerkin

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... 

Following the discretization of the solution domain Q (i.e. line AB) into two-node Lagrange elements, and representation of T as T = Ni(x)Ti) in terms of shape functions A, (.v), i = 1,2 within the space of a finite element Q, the elemental Galerkin-weighted residual statement of the differential equation is written as... 

Wulkow, M., 1996. The simulation of molecular weight distributions in polyreaction kinetics by discrete Galerkin methods. Macromol. Theory Simul. 5, 393-416. 

The unknown field quantity w, the test function, and all required derivatives of these quantities can be decomposed into contributions of the n elements following from the discretization of the system. By means of weighted basis functions (also denoted as ansatz functions) the following approximation is derived (Galerkin method) ... 

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 ... 

Based on the Galerkin weighted residual proach, the weak form of the discrete governing equation can be obtained... 

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