Computational Galerkin Methods by C. A. J. Fletcher

By C. A. J. Fletcher

In the wake of the pc revolution, a lot of it sounds as if uncon­ nected computational innovations have emerged. additionally, specific tools have assumed well known positions in yes components of program. Finite point tools, 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 circulate round plane wings and fuselages. those it sounds as if unrelated suggestions are firmly entrenched in laptop codes used on a daily basis by way of training scientists and engineers. lots of those scientists and engineers were drawn into the computational zone with out the convenience offormal computational education. frequently the formal computational education we do offer reinforces the arbitrary divisions among a few of the computational equipment on hand. one of many reasons of this monograph is to teach that many computational suggestions are, certainly, heavily similar. The Galerkin formula, that is getting used in lots of topic parts, offers the relationship. in the Galerkin frame-work we will be able to generate finite aspect, finite distinction, and spectral methods.

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4 that the Rayleigh-Ritz solution minimizing eq. 2) coincides with the Galerkin solution of eq. 1). Thus for the class of problems represented by eq. 1), the convergence properties of the Rayleigh-Ritz solution also apply to the Galerkin solution. For the Rayleigh-Ritz method (and hence the Galerkin method) Kantorovich and Krylov (1958) provide an estimate of the maximum error for an N-term trial solution of the self-adjoint ordinary differential equation ~(p:)-qy=! 11) in the region 0 :s;; x :s;; 1 with boundary conditions y(O) = y(l) = 0 and the subsidiary conditions p(x) > 0, q(x) ~ o.

Since eq. 11) is linear in u, boundary conditions of the form u = cp(s) on the boundary aD can always be reduced to homogeneous boundary conditions by changing the dependent variable u. Therefore we consider the homogeneous boundary-condition case here. Obtaining a solution to eq. 14) B. Ifeq. 13) is substituted into eq. 12) and iJI(ua)/iJa k = 0, the result is 2 + ~~ ~~k + fCPk) dxdy = 0. 15) Application of Green's theorem and noting the homogeneous boundary condition gives II (~:~a + ~:~a - f )CPkdXdY = 0, which is just the Galerkin method applied to eq.

Clearly this equivalence to a truncated Fourier series implies convergence in energy for the Galerkin solution as well. However although the approach of expanding in terms of energy-orthonormal functions is conceptually useful, the labor of creating the energy-orthonormal functions prevents this from being a practical scheme. 44 I. 5. Theoretical Properties For problems that can be cast in the equivalent variational form, the convergence properties associated with the Rayleigh-Ritz method carry over to the Galerkin method.

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