Galerkin approximation

As a matter of illustration, let us write in detail the numerical algorithm for computing ER(p,p ) with the PCM model (1.30) and (1.31) and the Galerkin approximation with P0 planar boundary elements ... 

For practical calculations, the integral over Y has to be discretized, which introduces an additional numerical error. An alternative consists in applying the Galerkin approximation to system (1.38), which is equivalent to Equation (1.37). The discretized apparent surface charge [cr] is obtained by solving successively the linear systems... 

It is understood that this equation must hold for all test functions, O, which must vanish at the boundaries where O = Oq. The Galerkin approximation

weak form solution O in Equation 23.11 can be expressed as ... 

The trial functions i = 0,1,..., N form a basis for an JV -F 1 dimensional space S. We define the Galerkin approximation to be that element < [Pg.374]

Unlike the traditional Taylor s series expansion method, the Galerkin approach utilizes basis functions, such as linear piecewise polynomials, to approximate the true solution. For example, the Galerkin approximation to the sample problem Equation 23.1 would require evaluating Equation 23.13 for the specific grid formation and specific choice of basis function ... 

In deriving the prediction eqnations for Cj (i), we make nse of the Galerkin approximation. Namely, the residue... 

Together with the Ritz-Galerkin approximation the coupled dynamic bending torsional differential beam equations can be solved. The linearized equation of motion of a mode numberj is obtained as a function of the generalized coordinate Y t) = q. (f),lightmodalstructuraldampinghas been added, see Reiterer et al. (2006)... 

Figure 8.1. Exact solution (o) and one- ( ) and two- (0) term Galerkin approximations to Equation 8.1.

The formulation of the finite element approximation starts with the Galerkin approximation, (aV) = —(Iy,), where <1> is our test function. We now use the finite element method to turn the continuous problems into a discrete formulation. First we discretize the solution domain, S2 = and define a finite dimensional subspace, V/, C V = < ) is continuous on... 

There is a close correspondence between Monte Carlo simulation and local polynomial approximations. The equations for Monte Carlo simulation can be obtained if instead of the Galerkin approximation, a collocation method is applied with respect to F, leading to... 

Now if the locations of the collocation points are the zeros of an orthogonal polynomial, the solution of Equation 2.23 approaches the Galerkin approximation [19], which has been shown to be the most accurate method of weighted... 

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