A numerical scheme based on Nédélec finite elements has been recently introduced to solve the eigenvalue problem for the curl operator in simply connected domains. This topological assumption is not just a technicality, since the eigenvalue problem is ill-posed on multiply connected domains, in the sense that its spectrum is the whole complex plane. However, additional constraints can be added to the eigenvalue problem in order to recover a well-posed problem with a discrete spectrum. Vanishing circulations on each non-bounding cycle in the complement of the domain have been chosen as additional constraints in this paper. A mixed weak formulation including a Lagrange multiplier (that turns out to vanish) is introduced and shown to be well-posed. This formulation is discretized by Nédélec elements, while standard finite elements are used for the Lagrange multiplier. Spectral convergence is proved as well as a priori error estimates. It is also shown how to implement this finite element discretization taking care of these additional constraints. Finally, a numerical test to assess the performance of the proposed methods is reported. 235 236 E. LARA, R. RODRÍGUEZ AND P. VENEGAS 2.A well posed eigenvalue problem on a multiply connected domain. Let Ω ⊂ R 3 be a bounded domain with a Lipschitz boundary Γ and outer unit normal n. We assume that Γ is either smooth or polyhedral.We restrict our attention to a multiply connected domain and we assume that there exist cutting surfaces Σ j , j = 1, . . . , J, such that the cut domain Ω 0 := Ω \ J j=1 Σ j becomes simply connected. More precisely, we assume that there exist a set {Σ j } J j=1of connected open subsets of smooth manifolds satisfying:• Σ j ⊂ Ω;• ∂Σ j ⊂ Γ;•Σ i ∩Σ j = ∅, i = j;• Ω 0 := Ω \ J j=1 Σ j is simply connected and pseudo-Lipschitz.
Voluntary shelter-in-place directives and lockdowns are the main nonpharmaceutical interventions that governments around the globe have used to contain the Covid-19 pandemic. In this paper, we study the impact of such interventions in the capital of a developing country, Santiago, Chile, that exhibits large socioeconomic inequality. A distinctive feature of our study is that we use granular geolocated mobile phone data to construct mobility measures that capture (1) shelter-in-place behavior and (2) trips within the city to destinations with potentially different risk profiles. Using panel data linear regression models, we first show that the impact of social distancing measures and lockdowns on mobility is highly heterogeneous and dependent on socioeconomic levels. More specifically, our estimates indicate that, although zones of high socioeconomic levels can exhibit reductions in mobility of around 50%–90% depending on the specific mobility metric used, these reductions are only 20%–50% for lower income communities. The large reductions in higher income communities are significantly driven by voluntary shelter-in-place behavior. Second, also using panel data methods, we show that our mobility measures are important predictors of infections: roughly, a 10% increase in mobility correlates with a 5% increase in the rate of infection. Our results suggest that mobility is an important factor explaining differences in infection rates between high- and low-incomes areas within the city. Further, they confirm the challenges of reducing mobility in lower income communities, where people generate their income from their daily work. To be effective, shelter-in-place restrictions in municipalities of low socioeconomic levels may need to be complemented by other supporting measures that enable their inhabitants to increase compliance. This paper was accepted by David Simchi Levi, healthcare management.
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