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Convergence for Eigenvalue Problems with Mesh Adaptivity

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Definition of the problem:

Throughout, will denote a bounded domain in ( or ). In fact will be assumed to be a polygon ( ) or polyhedron ( ). We will be concerned with the problem of finding an eigenvalue and eigenfunction satisfying

 for all (1.1)

where
 and (1.2)

Here, the matrix-valued function is required to be uniformly positive definite, i.e.
 for all with and all (1.3)

The methods which we describe below can be extended to piecewise smooth coefficients , but to reduce technical detail in the proofs we will assume in fact that is piecewise constant on and that the jumps in are aligned with the meshes (introduced below), for all . The scalar function is required to be bounded above and below by positive constants for all , i.e.
 for all (1.4)

Throughout the paper, for any polygonal (polyhedral) subdomain of , and any , and will denote the standard norm and seminorm in the Sobolev space . Also denotes the inner product. We also define the energy norm induced by the bilinear form :

for all
which, by (1.3), is equivalent to the seminorm. (The equivalence constant depends on the contrast , but we are not concerned with this dependence in the present paper.) We also introduce the weighted norm:
and note the norm equivalence
 (1.5)

Rewriting the eigenvalue problem (1.1) in standard normalised form, we seek such that

 (1.6)

By the continuity of and and the coercivity of on it is a standard result that (1.6) has a countable sequence of non-decreasing positive eigenvalues , with corresponding eigenfunctions .

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

Assumption 1.1 We assume that there exists a constant and with the following property. For , if solves the problem for all , then .
Assumption 1.1 is satisfied with when is constant (or smooth) and is convex. In a range of other practical cases , for example non-convex, or having a discoutinuity across an interior interface. Under Assumption 1.1 it follows that the eigenfunctions of the problem (1.6) satisfy .

To approximate problem (1.6) we use the continuous linear finite element method. Accordingly, let denote a family of conforming triangular ( ) or tetrahedral ( ) meshes on . Each mesh consists of elements denoted . We assume that for each , is a refinement of . For a typical element of any mesh, its diameter is denoted and the diameter of its largest inscribed ball is denoted . For each , let denote the piecewise constant mesh function on , whose value on each element is and let . Throughout we will assume that the family of meshes is shape regular, i.e. there exists a constant such that

 for all and all (1.7)

In the later sections of the paper the will be produced by an adaptive process which ensures shape regularity.

We let denote the usual finite dimensional subspace of , consisting of all continuous piecewise linear functions with respect to the mesh . Then the discrete formulation of problem (1.6) is to seek the eigenpairs such that

 (1.8)

The problem (1.8) has positive eigenvalues (counted according to multiplicity) which we denote in non-decreasing order as .