Mathematical Analysis

Download Acta Numerica 2010 (Volume 19) by Arieh Iserles PDF

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By Arieh Iserles

Acta Numerica is an annual e-book containing invited survey papers via best researchers in numerical arithmetic and clinical computing. The papers current overviews of contemporary advancements of their sector and supply 'state of the artwork' thoughts and research.

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Example text

The name ‘edge finite elements’ comes from the nature of the degrees of freedom which, for lowest-order approximation, are associated to moments along the edges of the triangulation. 1. First 50 discrete eigenvalues computed with piecewise linears on the unstructured mesh (N = 4, 8, 16). 2. Some eigenvalues computed with piecewise linears on the unstructured mesh for N = 16. 1 with lowest-order edge elements. 1. 3) is that the zero frequency is approximated by discrete values that are exactly equal to zero (up to machine precision).

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. The analysis will be performed with standard tools in the case of Dirichlet boundary conditions and piecewise linear finite elements, but can be applied with minor modifications to Neumann or mixed boundary conditions and to higherorder finite elements.

Let T ∗ : X → X denote the adjoint of T . Then λ ∈ σ(T ∗ ) if and only if λ ∈ σ(T ), where λ denotes the conjugate of λ. In particular, the eigenvalues of self-adjoint operators are real. The algebraic multiplicity of λ ∈ σ(T ∗ ) is equal to the algebraic multiplicity of λ ∈ σ(T ) and the ascent multiplicity of λ − T ∗ is equal to that of λ − T . 7. Variationally posed eigenvalue problems In this section we introduce some preliminary results on variationally posed eigenvalue problems. The main theoretical results are presented in Section 9.

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