Least-squares and streamline upwind Petrov-Galerkin SUPG schemes

2 Least-square.s and streamline upwind Petrov-Galerkin (SUPG) schemes 

The inconsistent streamline upwind scheme described in the last section is fonuulated in an ad hoc manner and does not correspond to a weighted residual statement in a strict sense. In tins seetion we consider the development of weighted residual schemes for the finite element solution of the energy equation. Using vector notation for simplicity the energy equation is written as 

After application of the 6 time-stepping method (see Chapter 2, Section 2.5) and following the procedure outlined in Chapter 2, Section 2.4, a functional representing the sum of the squares of the approximation error generated by the finite element discretization of Equation (4.118) is formulated as 

Assiuning constant density and specific heat (i.e. pc = constant) Equation (4.120) is written as 

If the second and third terms in the weight function are neglected the standard Galerkin scheme will be obtained. 

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