Acta Numerica 2010 (Volume 19) by Arieh Iserles

By Arieh Iserles

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

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1). 5. Our compactness assumption can be modified by assuming that T : V → V is compact. 7) with the natural modifications. 8), it is useful to recall that the discrete operator Th can be seen as Th = Ph T , where Ph : V → Vh is the elliptic projection associated to the bilinear form a. This fact is a standard consequence of Galerkin orthogonality and implies that T − Th can be written as (I − Ph )T , where I denotes the identity operator. The next proposition can be used to prove convergence in norm.

Eigenvalues computed with lowest-order RT elements on the unstructured mesh sequence of triangles. 6. 5. 0) 27 Eigenvalue problems RT elements are quite sensitive to the orientation of the mesh. 5. 6 It is interesting to note that in this case the eigenvalues may be approximated from above or below. Even the same eigenvalue can present numerical lower or upper bounds depending on the chosen mesh. 5. The Maxwell eigenvalue problem Maxwell’s eigenvalue problem can be written as follows by means of Amp`ere and Faraday’s laws: given a domain Ω ∈ R3 , find the resonance frequencies ω ∈ R3 (with ω = 0) and the electromagnetic fields (E, H) = (0, 0) such that curl E = iωµH in Ω, curl H = −iωǫE in Ω, E×n=0 on ∂Ω, H·n=0 on ∂Ω, where we assumed perfectly conducting boundary conditions, and ǫ and µ denote the dielectric permittivity and magnetic permeability, respectively.

6 can be used to show that, if T is compact from V into V , then a stronger pointwise convergence of Ph to the identity, from V into V , is sufficient to ensure the norm convergence T − Th L(V ) → 0 when h → 0. 7). 8. A direct proof of convergence for Laplace eigenvalues A fundamental example of elliptic partial differential equation is given by the Laplace operator. Although the convergence theory of the finite element approximation of Laplace eigenmodes is a particular case of the analysis presented in Sections 7 and 9, we now study this basic example.

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