A posteriori error estimates for the virtual element method

An a posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing

Building bridges connections and challenges in modern approaches to numerical partial differential equations

This volume contains contributed survey papers from the main speakers at the LMS/EPSRC Symposium “Building bridges: connections and challenges in modern approaches to numerical partial differential equations”. This meeting took place in July 8-16, 2014, and its main purpose was to gather specialists in emerging areas of numerical PDEs, and explore the connections between the different approaches. The type of contributions ranges from the theoretical foundations of these new techniques, to the applications of them, to new general frameworks and unified approaches that can cover one, or more than one, of these emerging techniques

Democrazia e cultura religiosa. Studi in onore di Pietro Scoppola

Il volume raccoglie saggi di storia contemporanea ispirati agli studi di Pietro Scoppola, le cui ricerche hanno spaziato su molti aspetti della vita religiosa e politica italiana tra Ottocento e Novecento. Attraverso un percorso culturale e scientifico che si è intrecciato con un costante impegno civile, istituzionale e politico, Scoppola ha manifestato particolare attenzione sia alle spinte di rinnovamento religioso sia alle istanze di riforma politica ed istituzionale, giungendo ad accostare in modo esplicito rinnovamento religioso e adesione ai valori democratici Il volume ripercorre molti dei temi affrontati da Scoppola, dal cattolicesimo liberale al modernismo, dai rapporti tra Chiesa e fascismo alla proposta politica di De Gasperi, dalla repubblica dei partiti alla “costituzione contesa”

Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants

In this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady Stokes flows in affinely parametrized geometries, focusing on the role played by the Brezzi’s and Babuška’s stability constants. The crucial ingredients of the methodology are a Galerkin projection onto a low-dimensional space of basis functions properly selected, an affine parametric dependence enabling to perform competitive Offline-Online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables. The combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction, ideally suited for real-time simulation and many-query contexts (e.g. optimization, control or parameter identification). In particular, in this work we focus on (i) the stability of the reduced basis approximation based on the Brezzi’s saddle point theory and the introduction of a supremizer operator on the pressure terms, (ii) a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the Babuška’s inf-sup constant (including residuals calculations), (iii) the computation of a lower bound of the stability constant, and (iv) different options for the reduced basis spaces construction. We present some illustrative results for both interior and external steady Stokes flows in parametrized geometries representing two parametrized classical Poiseuille and Couette flows, a channel contraction and a simple flow control problem around a curved obstacle.United States. Air Force Office of Scientific Research (Grant FA9550-07-1-0425)United States. Air Force Office of Scientific Research (Office of the Secretary of Defense Grant FA9550-09-1-0613