Mixed finite element methods in Elasticity---inf-sup stability vs stabilisation

Ramon Codina

Universitat Politècnica de Catalunya

Thursday 11th January 2024

14:00:00

Aula Consiglio VII piano - Dipartimento di Matematica

In this talk we consider the finite element approximation of some mixed formulations of linear elasticity, namely, the displacement-pressure, the displacement-stress and the displacement-pressure-stress approaches. As for any mixed formulation, involving unknowns belonging to different functional spaces, the global inf-sup stability that ensures well posedness of the problem is a consequence of “little” inf-sup conditions that need to be satisfied between the interpolating spaces of the different unknowns. Approximations satisfying these conditions are sometimes difficult to implement, and even very rare, as in the case of the displacement-pressure-stress approach. The alternative is to modify the discrete problem by adding stabilisation terms to the Galerkin equations that yield stable approximations for any choice of the interpolating spaces. We discuss this possibility and show that one may prove stability and convergence in the same norms as for the continuous problem. Contatto: simona.perotto@polimi.it

Ramon Codina is Professor of Continuum Mechanics and Structural Analysis at the Technical University of Catalonia (UPC), where he began his academic career in 1990. He teaches courses on continuum mechanics and mechanics of structures, as well as courses on computational mechanics and functional analysis in mechanics. His research is concerned with the development, implementation and numerical analysis of finite element methods applied to physical problems in engineering, mainly in fluid mechanics, but also in solids and electromagnetism. He is responsible for the research group Numerical Analysis and Computation, identified as a group of excellence by the Generalitat de Catalunya, and leader of the Fluid Mechanics Group at the International Centre for Numerical Methods in Engineering (CIMNE). In recognition to his research, he has received several awards, among which the Prandtl Medal from Eccomas (2018), the ICREA Academia Award (2011, 2016 and 2021) from the Catalan Government, the IACM Young Investigator Award (2004), the JL Lions Award to Young Scientists in Computational Mathematics from Eccomas (2000) and the Distinció de la Generalitat de Catalunya per a la Recerca Universitària (2000).