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Convergence for Eigenvalue Problems with Mesh Adaptivity

 

Convergence result:

The main result of this paper is Theorem 3.2 below which proves convergence of the adaptive method and also demonstrates the decay of oscillations of the sequence of approximate eigenfunctions. Before proving this result we need a final lemma.

Theorem 3.2 Provided the initial mesh $ \mathcal{T}_0$ is chosen so that $ H_0^\mathrm{max}$ is small enough, there exists a constant $ p \in (0,1)$ such that the recursive application of Algorithm 1 yields a convergent sequence of approximate eigenvalues and eigenvectors, with the property:
$\displaystyle \parallel\!\!\vert u-u_n\parallel\!\!\vert _{\Omega}\ \leq \ C_{0}qp^n,$ (3.1)

and
$\displaystyle \lambda_n \ \mathrm{osc}(u_n,\mathcal{T}_n) \ \leq \ C_1p^n,$ (3.2)

where $ C_0$ and $ C_1$ are positive constants.
Remark 3.3 The initial mesh convergence threshold and the constants $ C_1$ and $ C_2$ may depend on $ \theta$ , $ \tilde{\theta}$ and $ \lambda$ .

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