Spectral minimal partitions of a sector
Virginie Bonnaillie-Noël Corentin Léna
Discrete & Continuous Dynamical Systems - B 2014, 19(1): 27-53 doi: 10.3934/dcdsb.2014.19.27
In this article, we are interested in determining spectral minimal $k$-partitions for angular sectors. We first deal with the nodal cases for which we can determine explicitly the minimal partitions. Then, in the case where the minimal partitions are not nodal domains of eigenfunctions of the Dirichlet Laplacian, we analyze the possible topologies of these minimal partitions. We first exhibit symmetric minimal partitions by using a mixed Dirichlet-Neumann Laplacian and then use a double covering approach to catch non symmetric candidates. In this way, we improve the known estimates of the energy associated with the minimal partitions.
keywords: nodal domains numerical simulations Spectral theory finite element method. Aharonov-Bohm Hamiltonian minimal partitions
Harmonic oscillators with Neumann condition on the half-line
Virginie Bonnaillie-Noël
Communications on Pure & Applied Analysis 2012, 11(6): 2221-2237 doi: 10.3934/cpaa.2012.11.2221
We consider the spectrum of the family of one-dimensional self-adjoint operators $-{\mathrm{d}}^2/{\mathrm{d}}t^2+(t-\zeta)^2$, $\zeta\in \mathbb{R}$ on the half-line with Neumann boundary condition. It is well known that the first eigenvalue $\mu(\zeta)$ of this family of harmonic oscillators has a unique minimum when $\zeta\in\mathbb{R}$. This paper is devoted to the accurate computations of this minimum $\Theta_{0}$ and $\Phi(0)$ where $\Phi$ is the associated positive normalized eigenfunction. We propose an algorithm based on finite element method to determine this minimum and we give a sharp estimate of the numerical accuracy. We compare these results with a finite element method.
keywords: finite element method finite difference method Harmornic oscillator error estimate Eigenvalue

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