Working equations of the least-squares scheme in Cartesian coordinate systems

11 Working equations of the least-squares scheme in Cartesian coordinate systems 

Following the procedure described in Chapter 3, Section 1.6, after the discretization of Equations (4.1) and (4.4) the generated residuals are used to formulate a functional as 

It is evident that application of Green s theorem cannot eliminate second-order derivatives of the shape functions in the set of working equations of the least-sc[uares scheme. Therefore, direct application of these equations should, in general, be in conjunction with C continuous Hermite elements (Petera and Nassehi, 1993 Petera and Pittman, 1994). However, various techniques are available that make the use of elements in these schemes possible. For example, Bell and Surana (1994) developed a method in which the flow model equations are cast into a set of auxiliary first-order differentia] equations. They used this approach to construct a least-sciuares scheme for non-Newtonian flow equations based on equal-order C° continuous, p-version hierarchical elements. 

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