One key feature of isogeometric analysis is that it allows smooth shape functions. Indeed, when isogeometric spaces are constructed from p-degree splines (and extensions, such as NURBS), they enjoy up to C p−1 continuity within each patch. However, global continuity beyond C 0 on so-called multi-patch geometries poses some significant difficulties. In this work, we consider planar multipatch domains that have a parametrization which is only C 0 at the patch interface. On such domains we study the h-refinement of C 1 -continuous isogeometric spaces. These spaces in general do not have optimal approximation properties. The reason is that the C 1 -continuity condition easily over-constrains the solution which is, in the worst cases, fully locked to linears at the patch interface. However, recently [21] has given numerical evidence that optimal convergence occurs for bilinear two-patch geometries and cubic (or higher degree) C 1 splines. This is the starting point of our study. We introduce the class of analysis-suitable G 1 geometry parametrizations, which includes piecewise bilinear parametrizations. We then analyze the structure of C 1 isogeometric spaces over analysis-suitable G 1 parametrizations and, by theoretical results and numerical testing, discuss their approximation properties. We also consider examples of geometry parametrizations that are not analysis-suitable, showing that in this case optimal convergence of C 1 isogeometric spaces is prevented. arXiv:1509.07619v2 [math.NA]

We propose a population approach to model the beginning of the French COVID-19 epidemic at the regional level. We rely on an extended Susceptible-Exposed-Infectious-Recovered (SEIR) mechanistic model, a simplified representation of the average epidemic process. Combining several French public datasets on the early dynamics of the epidemic, we estimate region-specific key parameters conditionally on this mechanistic model through Stochastic Approximation Expectation Maximization (SAEM) optimization using Monolix software. We thus estimate basic reproductive numbers by region before isolation (between 2.4 and 3.1), the percentage of infected people over time (between 2.0 and 5.9% as of May 11 th , 2020) and the impact of nationwide lockdown on the infection rate (decreasing the transmission rate by 72% toward a R e ranging from 0.7 to 0.9). We conclude that a lifting of the lockdown should be accompanied by further interventions to avoid an epidemic rebound.