Structure-preserving schemes for the Maxwell system: conforming and non-conforming Finite Elements
Martin Campos Pinto, Laboratoire JL Lions CNRS & UPMC, Paris
Sala conferenze IMATI-CNR, Pavia - Tuesday, November 21, 2017 h.15:00
Abstract. Approximation and stability properties of mixed methods are known to be deeply connected with the compatibility relations satisfied by the discrete spaces, which usually reproduce at the discrete level some geometric structure of the functional spaces where the exact solutions live, such as De Rahm diagrams involving grad, curl and div operators. Several structure-preserving discretizations have been developed in the last decades, which involve so-called conforming Finite Element spaces that are embedded in the respective smoothness spaces, namely H^1, H(curl) or H(div).
This framework has been very successful in the stable discretization of problems with divergence constraints, such as the Maxwell equations.
In this talk I will present a method to extend these structure-preserving discretizations to discontinuous spaces, so as to design spectrally correct operators and long-time stable schemes for evolution problems with sources. In 2D some of the resulting schemes coincide with standard DG methods. In 3D our discretizations offer interesting alternatives to existing models, as their properties do not require penalty terms or divergence cleaning techniques.
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