By Arieh Iserles

Acta Numerica is an annual book containing invited survey papers via best researchers in numerical arithmetic and medical computing. The papers current overviews of modern advancements of their zone and supply 'state of the artwork' options and research.

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**Sample text**

Eigenvalues computed with lowest-order RT elements on the uniform mesh sequence of squares. 0) 4096 The RT space is used for the approximation of Σ. One of the main properties is that the finite element space consists of vector fields that are not globally continuous, but only conforming in H(div; Ω). This is achieved by requiring the normal component of the vector to be continuous across the elements, and the main tool for achieving this property is the so-called Piola transform, from the reference to the physical element.

11) Now comes a crucial point concerning the uniqueness of the normalized eigenfunctions. We have already observed that the normalization of the eigenfunctions does not identify them in a unique way (even in the case of simple eigenvalues), but only up to their sign. Here we have to choose the (k) appropriate sign of uh in order to have a good approximation of u(k) . 9) is satisfied. 10), we can conclude that, in the case of a simple eigenfunction u(k) , there (k) exists an appropriate choice of the sign of uh such that (k) u(k) − uh L2 (Ω) (k) ≤ 2(1 + ρh ) u(k) − Πh u(k) L2 (Ω) .

2. 10): find λh ∈ R and ph ∈ rot(Σh ) = Uh with ph ≡ 0 such that, for some σ h ∈ Σh , (σ h , τ ) + (ph , rot τ ) = 0 ∀τ ∈ Σh , (rot σ h , q) = −λh (ph , q) ∀q ∈ Uh . 6. The first spurious eigenfunction (centre) on the criss-cross mesh for N = 8. 7. The second and third spurious eigenfunctions (numbers 80 and 81) on the criss-cross mesh for N = 8. 36 D. 3. 9. 0) In particular, when Σh consists of edge or nodal elements (of lowest order), Uh is the space of piecewise constant functions with zero mean value.