By Guderley K. G., Keller C. L.
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Additional resources for A Basic Theorem in the Computation of Ellipsoidal Error Bounds
Numerical solutions for the linear hyperbolic problem at t = 2π for N = 16 for Fourier Galerkin and several ﬁnite-diﬀerence schemes In most practical applications the beneﬁt of the spectral method is not the extraordinary accuracy available for large N but rather the small size of N necessary for a moderately accurate solution. 2 A Chebyshev Collocation Method for the Heat Equation Fourier series, despite their simplicity and familiarity, are not always a good choice for the trial functions. In fact, for reasons that will be explored in the next chapter, Fourier series are only advisable for problems with periodic boundary conditions.
With a proper choice of the quadrature formulas, the ﬁnite series deﬁned by the discrete transform is actually the interpolant of u at the quadrature nodes. If the properties of accuracy (in particular the spectral accuracy) are retained by replacing the ﬁnite transform with the discrete transform, then the interpolant series can be used instead of the truncated series to approximate functions. , with an operation count with leading term (5/2)N log2 N , where N is the number of polynomials, rather than with the 2N 2 operations required by a matrix-vector multiplication.
26). 26) are based on the strong formulation of the diﬀerential equation, since the approximate solution is required to satisfy the diﬀerential equation exactly at a set of discrete points, in this case called the collocation points. One can formally obtain the same equations starting from a weak formulation of the problem by taking as test functions the (shifted) Dirac delta-functions (distributions) ψj (x) = δ(x − xj ) , j = 1, . . 29) and enforcing the conditions 1 −1 ∂uN − M(uN ) ψj (x) dx = 0 , ∂t j = 1, .
A Basic Theorem in the Computation of Ellipsoidal Error Bounds by Guderley K. G., Keller C. L.