Application of Greens theorem - weak formulations

Normally, the extra stress in the equation of motion is substituted in terms of velocity gradients and hence this equation includes second order derivatives of 

FINITE ELEMENT MODELLING OF POLYMERIC FLOW PROCESSES 

Therefore the second-order derivative of/ appearing in the original form of / is replaced by a term involving first-order derivatives of w and/plus a boundary term. The boundary terms are, normally, cancelled out through the assembly of the elemental stiffness equations over the common nodes on the shared interior element sides and only appear on the outside boundaries of the solution domain. However, as is shown later in this chapter, the appropriate treatment of these integrals along the outside boundaries of the flow domain depends on the prescribed boundary conditions. 

In practice, in order to maintain the symmetry of elemental coefficient matrices, some of the first order derivatives in the discretized equations may also be integrated by parts. 

The described application of Green s theorem which results in the derivation of the weak statements is an essential step in the formulation of robu.st U-V-P and penalty schemes for non-Newtonian flow problems. 

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