Inverse-type estimates on $hp$-finite element spaces and applications

This work is concerned with the development of inverse-type inequalities for piecewise polynomial functions and, in particular, functions belonging to $hp$-finite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite element functions.The inequalities are explicit both in the local polynomial degree and the local mesh size.The assumptions on the $hp$-finite element spaces are very weak, allowing anisotropic (shape-irregular) elements and varying polynomial degree across elements. Finally, the new inverse-type inequalities are used to derive bounds for the condition number of symmetric stiffness matrices of $hp$-boundary element method discretisations of integral equations, with element-wise discontinuous basis functions constructed via scaled tensor products of Legendre polynomials.