By Arieh Iserles

Acta Numerica is an annual booklet containing invited survey papers through best researchers in numerical arithmetic and clinical computing. The papers current overviews of modern advancements of their region and supply 'state of the artwork' concepts and research.

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First 50 discrete eigenvalues computed with piecewise bilinear elements on the uniform mesh of squares (N = 4, 8, 16). 3. 11) is no longer true. More precisely, the equivalence is true by choosing Uh = rot(Σh ), and Uh in this case is not a standard finite element space. For instance, in the case of lowest-order edge elements, Uh in each element K is made of functions C/|J|, where |J| is the determinant of the Jacobian of the mapping from ˆ to K, and C is a generic constant. 11), with Uh equal to the space of piecewise constant functions.

1999a). Unfortunately, this method does not seem to provide good results in the case of singular solutions (such as those obtained in an L-shaped domain). 7. Eigenvalues computed with the projected Q1 scheme (Q1 − P0 ) on the sequence of uniform meshes of squares with the spurious eigenvalue sorted in a diﬀerent way. 0) 44 D. Boffi PART TWO Galerkin approximation of compact eigenvalue problems This part of our survey contains the classical spectral approximation theory for compact operators. It is the core of our work, since all eigenvalue problems we are going to consider are related to compact operators, and we will constantly rely on the fundamental tools described in Section 9.

Boffi (k) in order to estimate the diﬀerence (u(k) − uh ) as follows: (k) u(k) − uh L2 (Ω) (k) (k) ≤ u(k) − Πh u(k) + Πh u(k) − wh (k) + wh − u h . 6) can be easily estimated in terms of powers of h using the properties of Πh ; let us start with the analysis of the second term. (k) From the definition of wh , we have (k) (i) Πh u(k) − wh = (i) (Πh u(k) , uh )uh , i=k which gives (k) 2 Πh u(k) − wh (i) (Πh u(k) , uh )2 . 7) i=k We have 1 (i) (Πh u(k) , uh ) = (i) (i) λh (grad(Πh u(k) ), grad uh ) 1 (i) (grad u(k) , grad uh ) = (i) = λ(k) (i) λh λh (i) (u(k) , uh ), that is, (i) (i) (i) λh (Πh u(k) , uh ) = λ(k) (u(k) , uh ).