Acta Numerica 2011: Volume 20 by Arieh Iserles

By Arieh Iserles

Acta Numerica is an annual ebook containing invited survey papers via prime researchers in numerical arithmetic and medical computing. The papers current overviews of contemporary advancements of their zone and supply 'state of the artwork' innovations and research.

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Allgower, B¨ohmer, Georg and Miranda (1992), Allgower, Georg, Miranda and Tausch (1998), Bossavit (1986), Douglas and Mandel (1992) and Georg and Miranda (1992). Our exposition, based on the group algebra, is explained in detail in ˚ Ahlander and Munthe-Kaas (2005). 4 The Laplacian can indeed be defined as the unique (up to constant) second-order linear differential operator which commutes with all isometries. Topics in structure-preserving discretization 35 G-equivariant matrices. A group is a set G with a binary operation g, h → gh, inverse g → g −1 and identity element e, such that g(ht) = (gh)t, eg = ge = g and gg −1 = g −1 g = e for all g, h, t ∈ G.

Z. Munthe-Kaas and B. Owren if a = g∈G a(g)g and b = a∗b= a(g)g ∗ b(h)h, then b(h)h g∈G where h∈G = h∈G g∈G a(gh−1 )b(h) = (a ∗ b)(g) = (a ∗ b)(g)g, a(g)b(h)(gh) = g,h∈G h∈G a(h)b(h−1 g). 4) h∈G Consider a G-equivariant A ∈ Cn×n in the case where G acts freely and transitively on I. In this case there is only one orbit of size |G| and hence I may be identified with G. , A(gh−1 ) = Agh−1 ,e = Ag,h . 5) Cn Similarly, any vector x ∈ corresponds uniquely to x = g∈G x(g)g ∈ CG, where x(g) = xg for all g ∈ G.

A route to spectral convergence is by a change of variables, which turns the trigonometric polynomials {cλ (t)} and {sλ (t)} into multivariate Chebyshev polynomials of the first and second kind. Before we dive into this topic, we remark on an alternative approach. Trigonometric polynomials with generalized symmetries. , a spanning set of vectors which is not linearly independent) generated by taking both the cosine and sine functions. In the frame we compute an approximating function by solving the following periodic extension problem.

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