By Arieh Iserles
Acta Numerica is an annual e-book containing invited survey papers by means of top researchers in numerical arithmetic and medical computing. The papers current overviews of contemporary advancements of their region and supply 'state of the paintings' thoughts and research.
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Extra resources for Acta Numerica 2010 (Volume 19)
The convergence analysis for the eigenvalues computed with this new element (sometimes referred to as the ABF element) can be found in Gardini (2005). Another, cheaper, cure consists in using a projection technique, which can also be interpreted as a reduced integration strategy (Boﬃ et al. 2006c). 9), or, equivalently, to using the midpoint rule in order to evaluate the integral (rot uh , rot v). 9. First 50 discrete eigenvalues computed with piecewise bilinear elements on the uniform mesh of squares (N = 4, 8, 16).
2) has solutions with zero frequency ω = 0. 3): in this case the eigenspace corresponding to the zero frequency is made of the harmonic vector fields plus the infinite-dimensional space grad(H01 (Ω)). It is well known that the space of harmonic vector fields is finite-dimensional, its dimension being the first Betti number of Ω. 3). 4b) (curl σ, q) = −λ(p, q) where H0 (div0 ; Ω) denotes the subspace of H0 (div; Ω) consisting of divergence-free vector fields and where the equivalence is given by λ = ω 2 , σ = u, and p = − curl σ/λ.
Then we consider the following approximating problem: find λh ∈ R and uh ∈ Vh , with uh = 0, such that (grad uh , grad v) = λh (uh , v) ∀v ∈ Vh . 2) We use the notation of the previous section for the eigensolutions of our continuous and discrete problems. In particular, we adopt the enumeration convention that eigenvalues are repeated according to their multiplicity. 2 that all eigenvalues are approximated from above, that is, (k) λ(k) ≤ λh ∀k, so that, in order to show the convergence of the eigenvalues, we need the upper bound (k) λh ≤ λ(k) + ε(h) with ε(h) tending to zero as h tends to zero.
Acta Numerica 2010 (Volume 19) by Arieh Iserles