Residuals weighted

Sum of squared, weighted residuals °F Number of degrees of freedom... 

Weighted Residual Finite Element Methods - an Outline... 

WEIGHTED RESIDUAL FINITE ELEMENT METHODS - AN OUTLINE... 

The weighted residual method provides a flexible mathematical framework for the construction of a variety of numerical solution schemes for the differential equations arising in engineering problems. In particular, as is shown in the followmg section, its application in conjunction with the finite element discretizations yields powerful solution algorithms for field problems. To outline this technique we consider a steady-state boundary value problem represented by the following mathematical model... 

The residual, Rq, is a function of position in Q. The weighted residual method is based on the elimination of this residual, in some overall manner, over the entire domain. To achieve this the residual is weighted by an appropriate number of position dependent functions and a summation is carried out. This is written as... 

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. 

Despite the simplicity of the outlined weighted residual method, its application to the solution of practical problems is not straightforward. The main difficulty arises from the lack of any systematic procedure that can be used to select appropriate basis and weight functions in a problem. The combination of finite element approximation procedures with weighted residual methods resolves this problem. This is explained briefly in the forthcoming section. 

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... 

At this stage the fonnulated Galerkin-weighted residual Equation (2.52) contains second-order derivatives. Therefore elements cannot generate an acceptable solution for this equation (using C elements the first derivative of... 

Development of weighted residual finite element schemes that can yield stable solutions for hyperbolic partial differential equations has been the subject of a considerable amount of research. The most successful outcome of these attempts is the development of the streamline upwinding technique by Brooks and Hughes (1982). The basic concept in the streamline upwinding is to modify the weighting function in the Galerkin scheme as... 

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... 

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