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High-precision eigenvalue bounds via Lehamnn-Goerisch method.
This project domonstrate the two-stage algorithm to provide high-precision guaranteed eigenvalue bounds for differential operators.
In the first stage, the non-conforming finite element methods are used to obtain lower eigenvalue bounds, which is regarded a generalization of the method proposed by the first author to handle the positive semi-definite bilinear forms in the eigenvalue problem formulation.
In the second stage, the Lehmann--Goerisch method along with high-order finite element methods and raw eigenvalue bounds are used to obtain high-precision eigenvalue bounds.
Two eigenvalue problems will be considered:
- The Laplacian eigenvalue problem
$$ -\Delta u = \lambda u \quad \text{ in } \Omega; \quad u=0 \text{ on } \partial \Omega~. $$
- The Steklov eigenvalue problem
$$ -\Delta u + u = 0 \quad \text{ in } \Omega; \quad \frac{\partial{u}}{\partial{n}}=\lambda u \text{ on } \partial \Omega~. $$
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