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Convergence for Eigenvalue Problems with Mesh Adaptivity

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Definition of the problem:

Throughout, $ \Omega$ will denote a bounded domain in $ \mathbb{R}^d$ ($ d = 2$ or $ 3$ ). In fact $ \Omega$ will be assumed to be a polygon ($ d = 2$ ) or polyhedron ($ d=3$ ). We will be concerned with the problem of finding an eigenvalue $ \lambda \in \mathbb{R}$ and eigenfunction $ 0\not= u \in H^1_0(\Omega)$satisfying

$\displaystyle a(u,v) : = \lambda \ b(u,v)\ ,$ for all $\displaystyle \quad v \in H^1_0(\Omega)\ ,$ (1.1)

where
$\displaystyle a(u,v) = \int_\Omega \nabla u(x) ^T \mathcal{A}(x) \nabla v(x) dx$ and $\displaystyle \quad b(u,v) = \int_\Omega \mathcal{B}(x) u(x) v(x) dx .$ (1.2)

Here, the matrix-valued function $ \mathcal{A}$ is required to be uniformly positive definite, i.e.
$\displaystyle \underline{a} \ \leq \ \xi^T \mathcal{A}(x) \xi \ \leq \ \overline{a}$ for all $\displaystyle \quad \xi \in \mathbb{R}^d$ with $\displaystyle \quad \vert\xi\vert=1$and all $\displaystyle \quad x \in \Omega.$ (1.3)

The methods which we describe below can be extended to piecewise smooth coefficients $ \mathcal{A}$ , but to reduce technical detail in the proofs we will assume in fact that $ \mathcal{A}$ is piecewise constant on $ \Omega$ and that the jumps in $ \mathcal{A}$ are aligned with the meshes $ \mathcal{T}_n$ (introduced below), for all $ n$ . The scalar function $ \mathcal{B}$ is required to be bounded above and below by positive constants for all $ x \in \Omega$ , i.e.
$\displaystyle \underline{b} \ \leq \ \mathcal{B}(x) \ \leq \ \overline{b}$ for all $\displaystyle \quad x \in \Omega.$ (1.4)

Throughout the paper, for any polygonal (polyhedral) subdomain of $ D \subset \Omega$ , and any $ s\in [0,1]$ , $ \Vert \cdot \Vert_{s,D}$ and $ \vert \cdot \vert_{s,D}$ will denote the standard norm and seminorm in the Sobolev space $ H^s(D)$ . Also $ (\cdot,\cdot)_{0,D}$ denotes the $ L_2(D)$ inner product. We also define the energy norm induced by the bilinear form $ a$ :

$\displaystyle \parallel\!\!\vert u\parallel\!\!\vert^2_\Omega:=a(u,u)$for all $\displaystyle u \in H^1_0(\Omega) \ , $
which, by (1.3), is equivalent to the $ H^1(\Omega)$ seminorm. (The equivalence constant depends on the contrast $ \overline{a}/\underline{a}$ , but we are not concerned with this dependence in the present paper.) We also introduce the $ L_2$ weighted norm:
$\displaystyle \Vert u \Vert_{0,\mathcal{B}, \Omega}^2 \ = \ b(u,u)\ = \ \int_\Omega \mathcal{B}(x) \vert u(x) \vert^2\,dx\ , $
and note the norm equivalence
$\displaystyle \sqrt{\underline{b}} \Vert v \Vert_{0,\Omega} \leq \Vert v \Vert_{0,\mathcal{B}, \Omega} \leq \sqrt{\overline{b}} \Vert v \Vert_{0,\Omega} \ .$ (1.5)

Rewriting the eigenvalue problem (1.1) in standard normalised form, we seek $ (\lambda,u) \in \mathbb{R}\times H^1_0(\Omega)$such that

$\displaystyle \left. \begin{array}{lcl} a(u,v) &=& \lambda \ b(u,v), \quad \t... ...ega)\\ \Vert u \Vert_{0,\mathcal{B},\Omega} &=& 1 \end{array}\quad \right\}$ (1.6)

By the continuity of $ a$ and $ b$ and the coercivity of $ a$ on $ H^1_0(\Omega)$ it is a standard result that (1.6) has a countable sequence of non-decreasing positive eigenvalues $ \lambda_j$ , $ j = 1,2,\ldots $ with corresponding eigenfunctions $ u_j \in H^1_0(\Omega)$.

We will need some additional regularity for the eigenfunctions $ u_j$ , which will be achieved by making the following regularity assumption for the elliptic problem induced by $ a$ :

Assumption 1.1 We assume that there exists a constant $ C_\mathrm{ell}>0$ and $ s\in [0,1]$ with the following property. For $ f \in L_2(\Omega)$ , if $ v\in H^1_0(\Omega)$ solves the problem $ a(v,w) = (f,w)_{0,\Omega} $ for all $ w \in H^1_0(\Omega)$ , then $ \Vert v \Vert_{{1+s},\Omega} \leq C_\mathrm{ell}\Vert f \Vert_{0, \Omega} $.
Assumption 1.1 is satisfied with $ s = 1$ when $ \mathcal{A}$ is constant (or smooth) and $ \Omega$ is convex. In a range of other practical cases $ s \in (0,1)$, for example $ \Omega$ non-convex, or $ \mathcal{A}$ having a discoutinuity across an interior interface. Under Assumption 1.1 it follows that the eigenfunctions $ u_j$ of the problem (1.6) satisfy $ \Vert u_j \Vert_{1+s,\Omega} \leq C_\mathrm{ell} \lambda_j\sqrt{\overline{b}}$.

To approximate problem (1.6) we use the continuous linear finite element method. Accordingly, let $ \mathcal{T}_n\ , n = 1,2,\ldots $denote a family of conforming triangular ($ d = 2$ ) or tetrahedral ($ d=3$ ) meshes on $ \Omega$ . Each mesh consists of elements denoted $ \tau \in \mathcal{T}_n$ . We assume that for each $ n$ , $ \mathcal{T}_{n+1}$ is a refinement of $ \mathcal{T}_n$ . For a typical element $ \tau$ of any mesh, its diameter is denoted $ H_\tau$ and the diameter of its largest inscribed ball is denoted $ \rho_\tau$ . For each $ n$ , let $ H_n$ denote the piecewise constant mesh function on $ \Omega$ , whose value on each element $ \tau \in \mathcal{T}_n$ is $ H_\tau$ and let $ H_n^{\max} = \max_{\tau\in \mathcal{T}_n} H_\tau$. Throughout we will assume that the family of meshes $ \mathcal{T}_n$ is shape regular, i.e. there exists a constant $ C_{\mathrm{reg}}$ such that

$\displaystyle H_\tau \leq C_{\mathrm{reg}} \rho_\tau,$ for all $\displaystyle \quad \tau \in \mathcal{T}_n$and all $\displaystyle \quad n = 1,2, \ldots \ .$ (1.7)

In the later sections of the paper the $ \mathcal{T}_n$ will be produced by an adaptive process which ensures shape regularity.

We let $ V_n$ denote the usual finite dimensional subspace of $ H^1_0(\Omega)$ , consisting of all continuous piecewise linear functions with respect to the mesh $ \mathcal{T}_n$ . Then the discrete formulation of problem (1.6) is to seek the eigenpairs $ (\lambda_n,u_n ) \in \mathbb{R}\times V_n$such that

$\displaystyle \left. \begin{array}{lcl} a(u_n,v_n) &=& \lambda_n \ b(u_n,v_n)... ... \Vert u_n \Vert_{0,\mathcal{B},\Omega} &=& 1\ . \end{array} \quad \right\}$ (1.8)

The problem (1.8) has $ N=\dim V_n$ positive eigenvalues (counted according to multiplicity) which we denote in non-decreasing order as $ \lambda_{n,1}\leq \lambda_{n,2} \leq \ldots \leq \lambda_{n,N} $.

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