EU Granted Project

Numerical Methods for PDE'S

Principal Investigator: Silvia Bertoluzza
Research Activities: Analisi numerica e calcolo scientifico

Year of start/end

2016 / 2021




Metodologie e tecniche d'ingagine:
We will tackle questions related to the properties of different classes of variational discretization methods defined on polygonal/polyhedral tessellations. In doing so we plan to track the influence of different mesh parameters, in order to understand how approximation depends on the geometric features the tessellation. Questions we would like to find an answer to are, e.g., (i) Are there better spaces than others (ii) Does the "best" space in the class depend on the problem, as we imagine?
Particular attention will be given to the robustness of the different methods and to the properties of the corresponding linear systems.
Other questions will be object of study, such as the one regarding the possibility to treat curved faces robustly, which are naturally generated by local modifications of the mesh and whose treatment need to be studied both from the theoretical point of view and from the algorithmic point of view, and the questions related to the design of adaptive methods.  
The subproject focuses on the development of a new mathematical approximation (and stability) theory for PEM methods, able to provide necessary and sufficient conditions for well posedness and accuracy, under minimal assumptions on the underlying polyhedral partition, thus extending as much as possible the class of tessellations that support stable and convergent methods (Analysis Suitable Polynomial Tessellations or ASPTs), possibly also including tessellation with curved bundaries. In particular we aim at identifying suitable minimal conditions that play the same role as "shape regularity" (in the sense of Ciarlet) in finite elements. Numerical experiments suggest that this condition should not be related to the number of faces or degeneracy of angles. We are confident that the class of ASPTs will allow for a tremendous flexibility.  

CNR (PDGP) Project

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