Weighted residual statement

Equation (2.45) represents the weighted residual statement of the original differential equation. Theoretically, this equation provides a system of m simultaneous linear equations, with coefficients Q , i = 1,... m, as unknowns, that can be solved to obtain the unknown coefficients in Equation (2.41). Therefore, the required approximation (i.e. the discrete solution) of the field variable becomes detemfined. 

Weighted residual statements in the context of finite element discretizations... 

As already discussed, variations of a field unknown within a finite element is approximated by the shape functions. Therefore finite element discretization provides a nat ural method for the construction of piecewise approximations for the unknown functions in problems formulated in a global domain. This is readily ascertained considering the mathematical model represented by Equation (2.40). After the discretization of Q into a mesh of finite elements weighted residual statement of Equ tion (2.40), within the space of a finite element T3<, is written as... 

In the standard Galerkin method (also called the Bubnov-Galerkin method) weight functions in the weighted residual statements are selected to be identical... 

In the earlier versions of the streamline upwinding scheme the modified weight function was only applied to the convection tenns (i.e. first-order derivatives in the hyperbolic equations) while all other terms were weighted in the usual manner. This is called selective or inconsistent upwinding. Selective upwinding can be interpreted as the introduction of an artificial diffusion in addition to the physical diffusion to the weighted residual statement of the differential equation. This improves the stability of the scheme but the accuracy of the solution declines. 

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... 

The weighted residual statement corresponding to Equation (2.80) is hence written as... 

The weighted residual statement of Equation (3.25) over an element domain Q,... 

Consider the weighted residual statement of the equation of motion in a steady state Stokes flow model, expressed as... 

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... 

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... 

Modified weighted residual statement = Original weighted residual statement +... 

Hence the integration of the weighted residual statement results in the forcing of the residuals to zero at specific points in the domain. 

Equation (2.7) and the derived equations (2.8) and (2.9) are the weighted residual statements for all common approximating techniques of the Laplace equation. 

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