Functional Inequalities for piecewise regular vector fields on general meshes

Michele Botti (Politecnico di Milano)

Abstract. In this talk, I will present several functional analysis tools that are instrumental to establish the stability and convergence analysis of nonconforming approximation methods on polyhedral meshes for a wide class of nonlinear problems. Starting from some preliminary results concerning broken Sobolev spaces, I will develop novel arguments to prove the broken version of Poincaré, Korn, trace inequalities and Sobolev embeddings. The main ingredient is a generalization of the local continuous trace inequality allowing to establish the results without restricting to piecewise polynomials or other discrete functional spaces and without requiring the definition of an interpolator mapping the discontinuous functions to continuous ones.