Galerkin weighted residual method

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... 

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

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... 

The unknown nodal displacements are obtained using Galerkin s weighted residual method. The inner product of the governing equation with respect to each of the interpolation functions is set to zero over the whole domain 2. However, the 4 order derivative term in governing equation requires the interpolation function to have continuity. In other words, the first derivatives of Nj with respect to x and z should be continuous along the inter-element boundary to avoid infinity in the integration of the so-called "stiffness" matrix. Hence, we introduce a new variable O such that... 

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

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),... 

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... 

It is seen from Table 8.1 that Galerkin appears to be the most accurate method for this specific problem. However, when more terms are used in the trial solution, the Galerkin method presents more analytical difficulties than the collocation method. As a matter of fact, all the weighted residual procedures... 

The FEM is part of a larger group of techniques that exploit the method of weighted residuals (MWR) [76]. These use a set of weighting functions to allow the approximation of a variable over a domain. The choice of weighting function leads to a number of different alternative formulations, including the collocation method, subdomain method, method of moments, and the Galerkin method. 

Method of weighted residuals - Petrov—Galerkin Method... 

A well known solution technique is the Method of Weighted Residuals also referred as Petrov-Galerkin method. Let Ym and Zm be finite dimensional subspaces of H spanned by yi,...,Pm and respectively. 

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