By C. A. J. Fletcher
In the wake of the pc revolution, various it seems that uncon nected computational ideas have emerged. additionally, specific tools have assumed widespread positions in sure parts of software. Finite point equipment, for instance, are used nearly solely for fixing structural difficulties; spectral tools have gotten the popular method of international atmospheric modelling and climate prediction; and using finite distinction tools is sort of common in predicting the circulation round airplane wings and fuselages. those it appears unrelated concepts are firmly entrenched in desktop codes used on a daily basis by way of working towards scientists and engineers. a lot of those scientists and engineers were drawn into the computational quarter with out the ease offormal computational education. frequently the formal computational education we do supply reinforces the arbitrary divisions among some of the computational equipment to be had. one of many reasons of this monograph is to teach that many computational ideas are, certainly, heavily comparable. The Galerkin formula, that's getting used in lots of topic components, presents the relationship. in the Galerkin frame-work we will be able to generate finite point, finite distinction, and spectral methods.
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Extra info for Computational Galerkin Methods
For eigenvalue problems, error estimates are more readily obtainable if the problem is amenable to the Rayleigh-Ritz method. Thus Kantorovich and Krylov (1958, p. 25) y(l) = O. For a trial functions 1= 1,2, ... 25) with q = 3/ 21100 1 in the region -I y( - I) = y(l) = 0 and a trial solution, Ya = (I - x 2)(a o + a 1x 2+ IAr) IT ~ x ~ + I and ... + a,_ 1x 2, - 2), Kantorovich and Krylov obtain error bounds that are extremely conservative. 00004 for 1= 3. Mikhlin (1964, p. 483) establishes that the Galerkin solution to the eigenvalue problem corresponding to eq.
The following trial solution was assumed: ua(x, t) = N I ek(t) sin kx. 31) possesses a unique solution and that the Galerkin solution Ua converges uniformly to the exact solution u. First derivatives with respect to x and t also converge uniformly, but d 2(ua)jdx2 converges in the mean to d 2ujdx 2. It was found that the L2 error satisfies Ilu a - ull~ ~ t2Fa2(t), where Fa == rr;~x It' [(L(ua)] 2 dxl 1/2. 33) Convergence for nonlinear initial-value problems is discussed by Finlayson (1972). In particular, the Navier~Stokes equations have received considerable attention (Ladyzhenskaya, 1963, 1975).
The Galerkin and subdomain methods have been compared by Fuller et al. (1970) for problems in space-dependent nuclear-reactor dynamics, and by Pomraning (1966) for a diffusion problem and an eigenvalue problem. The Galerkin method gave superior results to the subdomain method, particularly when eigenvalue problems were considered. This comparative section is concluded with the observation that the Galerkin method produces results of consistently high accuracy and has a breadth of application as wide as any method of weighted residuals.