A multigrid method for eigenvalue problems based on shifted-inverse power technique

In this paper, a type of full multigrid method is proposed to solve non-selfadjoint Steklov eigenvalue problems. Multigrid iterations for corresponding selfadjoint and positive definite boundary value problems generate proper iterate solutions that are subsequently added to the coarsest finite element space in order to improve approximate eigenpairs on the current mesh. Based on this full multigrid, we propose a new type of adaptive finite element method for non-selfadjoint Steklov eigenvalue problems. We prove that the computational work of these new schemes are almost optimal, the same as solving the corresponding positive definite selfadjoint boundary value problems. In this case, these type of iteration schemes certainly improve the overfull efficiency of solving the non-selfadjoint Steklov eigenvalue problem. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.

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An Algebraic Multigrid Method for Eigenvalue Problems and Its Numerical Tests

East Asian Journal on Applied Mathematics ◽

10.4208/eajam.210918.090519 ◽

2021 ◽

Vol 11 (1) ◽

pp. 1-19 ◽

Cited By ~ 1

Author(s):

Ning Zhang

Keyword(s):

Multigrid Method ◽

Eigenvalue Problems ◽

Algebraic Multigrid ◽

Algebraic Multigrid Method ◽

Numerical Tests

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PDE eigenvalue iterations with applications in two-dimensional photonic crystals

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2020014 ◽

2020 ◽

Vol 54 (5) ◽

pp. 1751-1776

Author(s):

Robert Altmann ◽

Marine Froidevaux

Keyword(s):

Photonic Crystals ◽

Eigenvalue Problem ◽

Adaptive Mesh Refinement ◽

Eigenvalue Problems ◽

A Priori ◽

Adaptive Mesh ◽

Inverse Power ◽

Two Dimensional ◽

Arnoldi Method ◽

The Matrix

We consider PDE eigenvalue problems as they occur in two-dimensional photonic crystal modeling. If the permittivity of the material is frequency-dependent, then the eigenvalue problem becomes nonlinear. In the lossless case, linearization techniques allow an equivalent reformulation as an extended but linear and Hermitian eigenvalue problem, which satisfies a Gårding inequality. For this, known iterative schemes for the matrix case such as the inverse power or the Arnoldi method are extended to the infinite-dimensional case. We prove convergence of the inverse power method on operator level and consider its combination with adaptive mesh refinement, leading to substantial computational speed-ups. For more general photonic crystals, which are described by the Drude–Lorentz model, we propose the direct application of a Newton-type iteration. Assuming some a priori knowledge on the eigenpair of interest, we prove local quadratic convergence of the method. Finally, numerical experiments confirm the theoretical findings of the paper.

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