By Arieh Iserles

Acta Numerica is an annual booklet containing invited survey papers via prime researchers in numerical arithmetic and medical computing. The papers current overviews of modern advancements of their sector and supply 'state of the artwork' suggestions and research.

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**Example text**

Munthe-Kaas and B. Owren The skew subspace E∧W does not form an algebra, but this can be corrected by dividing out the Weyl denominator. Deﬁne the Weyl vector ρ ∈ Λ as d λj = ρ= j=1 1 2 α. α∈Φ+ We deﬁne the Weyl denominator D ∈ E∧W as (−1)|w| ewρ . 16. , there exists a unique b ∈ E∨W such that a = bD. Proof. 2). Any a ∈ E∨W can be written as a polynomial in z1 , . . , zd , and hence the following polynomials are well-deﬁned. 17. For λ ∈ Λ we deﬁne multivariate Chebyshev polynomials of the ﬁrst and second kind, Tλ and Uλ , as the unique polynomials that satisfy Tλ (z1 , .

Wk ] which is consistent and cyclic, and a skew-symmetric map Dd depending on at most k − 1 arguments. The scheme is then δHd Un+k − Un = kDd . k∆t δ(Un , . . 34) This scheme conserves the function Hd in the sense that Hd [Un+1 , . . , Un+k ] = Hd [U0 , . . , Uk−1 ], n ≥ 0. 4. Spatial symmetries, high-order discretizations and fast group-theoretic algorithms In this chapter we survey some group-theoretic techniques applied to the discretization and solution of PDEs. 1 present algorithms for computing matrix and operator exponentials based on Fourier transforms on ﬁnite groups.

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. Find a smooth extension of f to L2 (T) and approximate this by the frame on . A detailed discussion of such techniques is found in Huybrechs (2010). One question is whether a frame consisting of only cλ (t) and sλ (t) might also be suﬃcient to obtain spectral convergence for higher-dimensional domains, or whether we have to include basis functions with other kinds of symmetries.