Working equations of the U-V-P scheme in Cartesian coordinate systems

4 Working equations of the U-V-P scheme in Cartesian coordinate systems 

Following the procedure described in Chapter 3, Section 1.1 the Galerkin-weighted residual statements corresponding to Equations (4.4) and (4.1) are written as 

In Equation (4.12) the discretization of velocity and pressure is based on different shape functions (i.e. NjJ = l,n and Mil= l,m where, in general, m n). The weight function used in the continuity equation is selected as -Mi to retain the symmetry of the discretized equations. After application of Green s theorem to the second-order velocity derivatives (to reduce inter-element continuity requirement) and the pressure terms (to maintain the consistency of the formulation) and algebraic manipulations the working equations of the U-V-P scheme are obtained as 

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