Galerkin-weighted residual

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... 

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... 

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... 

After rearranging, the Galerkin-weighted residual stateinent arising from Equation (5.12) can be written as... 

The next step will be to formulate the Galerkin-weighted residual for each of the governing equations... 

The nodal unknows a are to be chosen so as to satisfy the governing equations in an integral sense this can be done by using a Galerkin weighted residual formulation of the conservation equations for momentum and energy transport ... 

The Galerkin-weighted residual equations are obtained by inserting the test function into the SDT, with the boundary term deleted ... 

The Galerkin weighted residual method is employed to formulate the finite element discretisation. An implicit mid-interval backward difference algorithm is implemented to achieve temporal discretisation. With appropriate initial and boundary conditions the set of non-linear coupled governing differential equations can be solved. 

Application of the Galerkin weighted-residual method to Eq. 8.9 gives... 

Based on the Galerkin weighted residual proach, the weak form of the discrete governing equation can be obtained... 

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

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... 

The standard least-squares approach provides an alternative to the Galerkin method in the development of finite element solution schemes for differential equations. However, it can also be shown to belong to the class of weighted residual techniques (Zienkiewicz and Morgan, 1983). In the least-squares finite element method the sum of the squares of the residuals, generated via the substitution of the unknown functions by finite element approximations, is formed and subsequently minimized to obtain the working equations of the scheme. The procedure can be illustrated by the following example, consider... 

The two most common of the methods of weighted residuals are the Galerkin method and collocation. In the Galerkin method, the weighting functions are chosen to be the trial functions, which must be selected as members of a complete set of functions. (A set of functions is complete if any function of a given class can be expanded in terms of the set.) Also according to Finlayson (1972),... 

Fourier-Galerkin method. To illustrate the weighted residuals method we chose a Fourier-Galerkin method to solve a PDE of the form,... 

For a Galerkin solution, the weighting functions will be the same basis functions and the weighted residual expression will reduce to... 

We can now substitute the approximated linear function of the held variable into the governing eqn. (9.51), and form the weighted residual Galerkin finite element expression... 

Next, we apply Galerkin s weighted residual method and reduce the order of integration of the various terms in the above equations using the Green-Gauss Theorem (9.1.2) for each element. For a simpler presentation we will deal with each term in the above equations separately. The terms of the x-component (eqn. (9.95)) of the penalty formulation momentum balance become... 

Adopting the Galerkin method as a particular form of weighted residuals, i.e., considering the weights W, to be the same as the trial functions N after standard transformations of integrals in the relation (11), the next system of the ordinary differential equations with respect to nodal concentrations Q(t) may be derived ... 

Other methods may be more appropriate for equations with particular mathematical characteristics or when more accurate, robust, stable and efficient solutions are required. The alternative spectral methods can be classified as sub-groups of the general approximation technique for solving differential equations named the method of weighted residuals (MWR) [51]. The relevant spectral methods are called the collocation Galerkin, Tan- and Least squares methods. These methods can also be applied to subdomains. The subdomain... 

In the early 1970s, the standard finite element approximations were based upon the Galerkin formulation of the method of weighted residuals. This technique did emerge as a powerful numerical procedure for solving elliptic boundary value problems [102, 75, 53, 84, 50, 89, 17, 35]. The Galerkin finite element methods are preferable for solving Laplace-, Poisson- and and diffusion equations because they do not require that a variational principle exists for the problem to be analyzed. However, the power of the method is still best utilized in systems for which a variational principle exists, and it... 

---