By Bertrand Mercier
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Additional info for An Introduction to the Numerical Analysis of Spectral Methods
P C~(1) is dense in u e HI(1). D. ) u e Hi(l). 1. 2: the functions If, instead of SN (e inx), -N+I ~ n < N, we properties for the projection operator SN is of dimension we consider the space have some PN :L2 + SN" analogous SN spanned by approximation (We note that the space 2N, and the space SN is of dimension 2N+I). 32 7. First-Order Let L Eqtmtlons - The Galerkin Method be the first-order operator defined by au a(au) Lu ~ a ~ x + 'ax where a e C~(1) P is regular and periodic (real). We observe first that (Lu,v) = ~ - L is skew symmetric: ~ x + ~--x-----jvcx = ~ --~ for u,v ~ D(L) d~f H~(1).
Noting that t f d W~(s) = -W~(t-s) we have t (~N(S),U(s))ds = f ((WN(t_s),u~
An Introduction to the Numerical Analysis of Spectral Methods by Bertrand Mercier