21 research outputs found

    hp-adaptive Galerkin Time Stepping Methods for Nonlinear Initial Value Problems

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    This work is concerned with the derivation of an a posteriori error estimator for Galerkin approximations to nonlinear initial value problems with an emphasis on finite-time existence in the context of blow-up. The structure of the derived estimator leads naturally to the development of both h and hp versions of an adaptive algorithm designed to approximate the blow-up time. The adaptive algorithms are then applied in a series of numerical experiments, and the rate of convergence to the blow-up time is investigated

    Error control for time-splitting spectral approximations of the semiclassical Schrödinger equation

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    We prove a posteriori error estimates of optimal order in the L ∞(L2)-norm for time-splitting spectral methods applied to the linear Schrödinger equation in the semiclassical regime. The a posteriori error estimates are obtained by considering an appropriate extension in time of the numerical schemes and using energy techniques. Numerical experiments are presented that confirm our theoretical results.</p

    Participatory design to support students' web-based inquiry of complex, socio-scientific problems

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    This poster presents five, multilingual, web-based learning environments (LEs) that have been designed using the learning and teaching platform of STOCHASMOS as part of the CoReflect FP7 project. CoReflect brought together eight diverse and multi-disciplinary teams from six European states and one associated country to collaborate for the iterative design, enactment and research of problem-based innovative inquiry LEs. Using STOCHASMOS, each web-based LE was developed by a Local Working Group (LWG), which consisted of researchers, designers, scientists, and practicing teachers to address the research-practice gap. Each LE was piloted in authentic classroom conditions at the local level and revised based on design-based research principles. Subsequently, the LWGs translated their LEs to English. This allowed for sharing their LE with a collaborating LWG (cLWG) in another partner country. The LE was then translated in the cLWG local language and enacted in a different cultural context. The LEs focus on socio-scientific issues and target primary and secondary school students. Five out of seven are presented here: "Biotechnology" and "Climate Change" - Cyprus, "Nicotine Addiction" - Israel, "Extraterrestrial Life" - Sweden, and "Fog control" - Greece. They are available in English and in two additional languages each, for widespread use by primary and secondary school educators. Empirical, quantitative and qualitative results on the effectiveness of these five LEs in authentic classrooms are presented

    A posteriori error control and adaptivity for Crank-Nicolson finite element approximations for the linear Schrödinger equation

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    We derive optimal order a posteriori error estimates for fully discrete approximations of linear Schrödinger-type equations, in the L∞ (L2)-norm. For the discretization in time we use the Crank–Nicolson method, while for the space discretization we use finite element spaces that are allowed to change in time. The derivation of the estimators is based on a novel elliptic reconstruction that leads to estimates which reflect the physical properties of Schrödinger equations. The final estimates are obtained using energy techniques and residual-type estimators. Various numerical experiments for the one-dimensional linear Schrödinger equation in the semiclassical regime, verify and complement our theoretical results. The numerical implementations are performed with both uniform partitions and adaptivity in time and space. For adaptivity, we further develop and analyze an existing time-space adaptive algorithm to the cases of Schrödinger equations. The adaptive algorithm reduces the computational cost substantially and provides efficient error control for the solution and the observables of the problem, especially for small values of the Planck constant.</p

    Increasing International Capacity for CSCL: CoReflect, an example of sharing and adapting CSCL environments across europe

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    We report on differences in collaborative processes that took place when CSCL environments originally developed in one European country were adapted and enacted in another European country as part of the Digital Support for Inquiry, Collaboration, and Reflection on Socio-Scientific Debates (CoReflect) project. We describe what differences were designed and emerged as part of the adaptation process, and discuss implications for principles for cross-national adaptation of CSCL environments
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